Free Math Equation OCR. i2OCR is a free online Optical Character Recognition (OCR) that extracts Math Equation text from images and scanned documents so that it can be edited, formatted, indexed, searched, or translated. 100+ Recognition Languages. Multi Column Document Analysis. 100% FREE, Unlimited Uploads, No Registration Read More ...Are you tired of spending hours trying to solve complex equations manually? Look no further. The HP 50g calculator is here to make your life easier with its powerful Equation Libra...There is another concept related to complex number that is based on the following bit of algebra. (a + bi)(a − bi) = a2 − (bi)2 = a2 − b2i2 = a2 + b2. The complex number a − bi is called the complex conjugate of a + bi. If we let z = a + bi, we denote the complex conjugate of z as ˉz. So ˉz = ¯ a + bi = a − bi.23 Mar 2023 ... Remember, learning complex math equations takes time, effort, and patience. Be persistent, stay positive, and keep practicing until you feel ...There is another concept related to complex number that is based on the following bit of algebra. (a + bi)(a − bi) = a2 − (bi)2 = a2 − b2i2 = a2 + b2. The complex number a − bi is called the complex conjugate of a + bi. If we let z = a + bi, we denote the complex conjugate of z as ˉz. So ˉz = ¯ a + bi = a − bi.Solving math problems has often been stressful for students, engineers, and teachers. However, there are few math software that helps solve and explain complex mathematical equations accurately and efficiently. There are dozens of Math problem solving programs available online with different specializations and problem-solving …Private Site. Build a website. Sell your stuff. Write a blog. And so much more. Log in Start your website.Sep 9, 2019 · Two more answers for a complex math problem have been found.; Called the "summing of three cubes," the challenge is to find x, y, and z.It took over a million computing hours to find the solution. RealCalc is a scientific calculator app for Android users. It has over 20 million downloads and various 5-star reviews. The calculator app is built in a way that looks like a real calculator. RealCalc solves complex mathematical problems like powers, roots, exponents, trigonometric functions, and even logs.30 Sept 2016 ... Hi Bubble Community, I'm creating an exam app and will be featuring questions with complex math equations (Fractions, Integrals, ...Are you tired of spending hours solving complex math problems manually? Look no further than the HP 50g Equation Library. The HP 50g is a graphing calculator renowned for its exten...In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, ... complex numbers). An algebraic equation is univariate if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate (multiple variables, x, y, z ...The Fourier transform is an analysisprocess, decomposing a complex-valued function f(x){\displaystyle \textstyle f(x)}into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates f(x){\displaystyle \textstyle …Learn how to manipulate and solve equations involving complex numbers, which are numbers with real and imaginary parts. See examples of complex math …Graph paper is a versatile tool that has been used for centuries in the fields of math and science. Its grid-like structure makes it an essential tool for visualizing data, plottin...This is a list of unusually long mathematical proofs.Such proofs often use computational proof methods and may be considered non-surveyable.. As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would …Aug 14, 2021 · Complex functions. Let S S be a set of complex numbers. A function f f defined on S S is a rule that assigns to each z z in S S a complex number w w. The number w w is called the value of f f at z z and is denoted by f(z) f ( z); that is, w = f(z) w = f ( z). The set S S is called the domain of definition of f f. To get the complex numbers, we do a similar thing. Take the real numbers and add in 1. Every real number is complex. 2. There is a complex number i such that i²= -1. 3. The sum of two complex numbers is complex. 4. The product of two complex numbers is complex. 5. For any two complex numbers a and b, a^b is complex. With Tenor, maker of GIF Keyboard, add popular Math Equation animated GIFs to your conversations. Share the best GIFs now >>> Browse 758 complex math equation photos and images available, or start a new search to explore more photos and images. Math equations written on a blackboard. Physics teacher writing math equations on a blackboard. professor writing on the board while having a chalk and blackboard lecture (shallow DOF; color toned image) 19 Mar 2015 ... ... complex math equation using something like the Quadratic Formula? I could certainly type out something that looks like this: Solve x^2 + 3x ...With Tenor, maker of GIF Keyboard, add popular Math Equation animated GIFs to your conversations. Share the best GIFs now >>>Learn how to add, subtract, multiply, divide, and classify complex numbers, as well as how to plot them on the complex plane and use polar coordinates. Explore the properties of …Add a comment. 1. Since the output from math OCR for complex formulas will likely have bugs -- even humans have trouble with it -- you will have to proofread th results, at least if they matter. The (human) proofreader will then have to correct the results, meaning you need to have a math formula editor.RealCalc is a scientific calculator app for Android users. It has over 20 million downloads and various 5-star reviews. The calculator app is built in a way that looks like a real calculator. RealCalc solves complex mathematical problems like powers, roots, exponents, trigonometric functions, and even logs.21 Jan 2023 ... Comments11 ; How To Solve A Radical Differential Equation · 183 views ; Let's Simplify A Radical · 1.4K views ; Let's Compare Two Exponentials...The web page explains that the most complicated equation is the Standard Model Lagrangian, which describes the dynamics of every kind of particle and all …Calculators have become an essential tool for students, professionals, and even everyday individuals. Whether you need to solve complex equations or perform simple arithmetic calcu...Oct 6, 2021 · Multiply the numerator and denominator of a fraction by the complex conjugate of the denominator and then simplify. Ensure that any complex number is written in terms of the imaginary unit i. i. before performing any operations. Exercise 5.7.4. Rewrite in terms of imaginary unit i. √− 81. − 81 − − − − √. 9 Oct 2015 ... Numo is designed to help students solve complex math equations through the act of teaching. To use the app, students simply take a picture of ... A simple formula is a mathematical expression with one operator, such as 7+9. A complex formula has more than one mathematical operator, such as 5+2*8. When there is more than one operation in a formula, the order of operations tells your spreadsheet which operation to calculate first. In order to use complex formulas, you will need to ... Works across all devices. Use our algebra calculator at home with the MathPapa website, or on the go with MathPapa mobile app. Download mobile versions. Great app! Just punch in your equation and it calculates the answer. Not only that, this app also gives you a step by step explanation on how to reach the answer!The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. Khan Academy's Algebra 2 course is …Python program to solve quadratic equation. A quadratic equation is a polynomial equation of degree 2, which means it contains a term with a variable raised to the power of 2. It takes the form: ax2 + bx + c = 0. where, a, b, and c are coefficient and real numbers and also a ≠ 0. If a is equal to 0 that equation is not valid quadratic equation.To type an equation from scratch, press Alt += on your keyboard. or. Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. See the ribbon for more Structures and Convert options. Add an equation to the equation gallery. Select the equation you want to add.Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websit... In this chapter, you will learn the basics of complex integration, a powerful tool for studying complex functions and their properties. You will see how to define and compute complex line integrals, how to apply the Cauchy-Goursat theorem and the Cauchy integral formula, and how to use residues and the residue theorem to evaluate real integrals. This book also offers visual and interactive ... This online mathematical keyboard is limited to what can be achieved with Unicode characters. This means, for example, that you cannot put one symbol over another. While this is a serious limitation, multi-level formulas are not always needed and even when they are needed, proper math symbols still look better than improvised ASCII approximations.Select Insert > Equation or press Alt + =. To use a built-in formula, select Design > Equation. To create your own, select Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. Select Insert to bring your equation into the file. See the ribbon for more Structures and Convert options.Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as “the most beautiful equation.”. It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures (opens ….CMI was founded in 1998 by American businessman Landon T. Clay “to increase and disseminate mathematical knowledge.”. The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills …Mar 23, 2023 · Damn . . . . . . ChatGPT now uses Wolfram Alpha to deal with math as well as other factual information. Wolfram is offered as a plugin and not used by default to solve mathematical equations with ChatGPT. As far as we know, GPT-4's core capabilities are still based mainly on a Large Language Model (LLM). More complex formulas can contain several mathematical operators, such as 5+2*8. When there's more than one operation in a formula, the order of operations ...The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). When a single letter z=x+iy is used to denote a complex number, it is sometimes called an "affix." In component notation, z=x+iy can be written (x,y). The field of complex numbers …Find Complex Math Equations stock photos and editorial news pictures from Getty Images. Select from premium Complex Math Equations of the highest quality.Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Browse 758 complex math equation photos and images available, or start a new search to explore more photos and images. Math equations written on a blackboard. Physics teacher writing math equations on a blackboard. professor writing on the board while having a chalk and blackboard lecture (shallow DOF; color toned image) To type an equation from scratch, press Alt += on your keyboard. or. Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. See the ribbon for more Structures and Convert options. Add an equation to the equation gallery. Select the equation you want to add. To get the complex numbers, we do a similar thing. Take the real numbers and add in 1. Every real number is complex. 2. There is a complex number i such that i²= -1. 3. The sum of two complex numbers is complex. 4. The product of two complex numbers is complex. 5. For any two complex numbers a and b, a^b is complex. Jan 2, 2021 · There is another concept related to complex number that is based on the following bit of algebra. (a + bi)(a − bi) = a2 − (bi)2 = a2 − b2i2 = a2 + b2. The complex number a − bi is called the complex conjugate of a + bi. If we let z = a + bi, we denote the complex conjugate of z as ˉz. So ˉz = ¯ a + bi = a − bi. In summary, individuals discussed different mathematical equations that could potentially equal 25, as it is the age of the person in ...Oct 6, 2021 · Multiply the numerator and denominator of a fraction by the complex conjugate of the denominator and then simplify. Ensure that any complex number is written in terms of the imaginary unit i. i. before performing any operations. Exercise 5.7.4. Rewrite in terms of imaginary unit i. √− 81. − 81 − − − − √. The letter z is often used for a complex number: z = a + bi. z is a Complex Number. a and b are Real Numbers. i is the unit imaginary number = √−1. we refer to the real part and imaginary part using Re and Im like this: Re (z) = a, Im (z) = b. The conjugate (it changes the sign in the middle) of z is shown with a star: z * = a − bi.A complex number is the sum (or difference) of a real number and an imaginary number (that is, a number that contains the number i ). If a and b are regular numbers, then a + bi is a complex number. Complex numbers are "binomials" of a sort, and are added, subtracted, and multiplied in a similar way. (Division, which is further down the page ...With Tenor, maker of GIF Keyboard, add popular Math Equation animated GIFs to your conversations. Share the best GIFs now >>>RealCalc is a scientific calculator app for Android users. It has over 20 million downloads and various 5-star reviews. The calculator app is built in a way that looks like a real calculator. RealCalc solves complex mathematical problems like powers, roots, exponents, trigonometric functions, and even logs.There are a few ways to write Math equations in HTML: Use HTML Math symbols and tags to build simple equations. Refer to this list for the available Math symbols.; For example, ∛ 27 = 3 will display as ∛ 27 = 3. For complicated equations, use a library to help – Check out MathJax.; Lastly, use online generators to create … Rolle's Theorem Example. Face Value Example. Matrix Addition. Table Of 6. Cube Root Table. Algebraic operations on complex numbers such as addition, subtraction, multiplication, division are explained here in detail. Visit BYJU'S to learn more about algebraic operations on complex numbers. The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. Khan Academy's Algebra 2 course is …Download this stock image: Complex mathematical equation - GJ9JRF from Alamy's library of millions of high resolution stock photos, illustrations and ...To draw an equation, do the following: Enter a blank equation control by pressing Alt and =. From the contextual Equations ribbon, click Ink Equation in the Tools group ( Figure F ). Draw your ...18 Nov 2009 ... That one is pretty lackluster though. I'm looking forward to seeing what the math wizards here can come up with.30 Sept 2016 ... Hi Bubble Community, I'm creating an exam app and will be featuring questions with complex math equations (Fractions, Integrals, ... In this chapter, you will learn the basics of complex integration, a powerful tool for studying complex functions and their properties. You will see how to define and compute complex line integrals, how to apply the Cauchy-Goursat theorem and the Cauchy integral formula, and how to use residues and the residue theorem to evaluate real integrals. This book also offers visual and interactive ... In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, ... or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent ...The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. Khan Academy's Algebra 2 course is …Complex Numbers. When we combine a Real Number and an Imaginary Number we get a Complex Number: Examples: 1 + i. 39 + 3i. 0.8 − 2.2i. −2 + π i. √2 + i/2. Can a Number …You've got a couple problems. (1) You're missing parentheses for the denominator of form2 (and R has no way to know that you want to add a constant a in the denominator, or where to put any of the parameters, really), and much more problematic: (2) your 2nd model isn't linear, so lm won't work.. Fixing (1) is easy: form2 <- h ~ 1.3 + …The complex plane plays an important role in Mathematics. It is also known as z-plane, which is composed of two mutually perpendicular lines called axes.The difference between an expression and an equation is that an expression is a mathematical phrase representing a single value whereas an equation is a mathematical sentence asser...Solve complex equations step-by-step with this online calculator. Enter your equation and get detailed solutions, graphs, and explanations for any level of math.Quadratic Equations: Very Difficult Problems with Solutions. Problem 1. Solve the equation \displaystyle \frac {5} {2-x}+\frac {x-5} {x+2}+\frac {3x+8} {x^2-4}=0 2−x5 + x+2x−5 + x2 −43x+8 = 0. In the answer box, write the roots separated by a comma. Problem 2. If \displaystyle x^2-2ax+a^2=0 x2 −2ax+a2 = 0, find the value of ...30 Sept 2016 ... Hi Bubble Community, I'm creating an exam app and will be featuring questions with complex math equations (Fractions, Integrals, ...The Complex Number Calculator solves complex equations and gives real and imaginary solutions. Step 2: Click the blue arrow to submit. Choose "Find All Complex ...The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. Khan Academy's Algebra 2 course is …Nov 6, 2012 · The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk. Examples are illustrated to elucidate the solution procedure including the space-time fractional differential equation in complex domain, singular problems and Cauchy problems. Here, we consider ... Complex Analysis Complex analysis is the field of mathematics dealing with the study of complex numbers and functions of a complex variable. Wolfram|Alpha's authoritative computational ability allows you to perform complex arithmetic, analyze and compute properties of complex functions and apply the methods of complex analysis to solve …Select Insert > Equation or press Alt + =. To use a built-in formula, select Design > Equation. To create your own, select Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. Select Insert to bring your equation into the file. See the ribbon for more Structures and Convert options.Solving a complex algebraic formula with java. I am trying to solve for y in the following equation using Java: For the sake of visibility I divided the numerator and denominator into separate variables. I need to compute for x = -3 through x = 4 moving in increments of 0.5. // Now we compute the formula for all values in between -3 and 4 in ...The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). When a single letter z=x+iy is used to denote a complex number, it is sometimes called an "affix." In component notation, z=x+iy can be written (x,y). The field of complex numbers … Complex Numbers. A complex number is a number that can be written in the form a + bi a+bi, where a a and b b are real numbers and i i is the imaginary unit defined by i^2 = -1 i2 = −1. The set of complex numbers, denoted by \mathbb {C} C, includes the set of real numbers \left ( \mathbb {R} \right) (R) and the set of pure imaginary numbers. Explanation: For two imaginary numbers to be equal to each other, their imaginary parts must be equal. Therefore, we set, and solve for y in: 3y + 4 = 7. 3y = 3. y = 1. Report an …This video explains how to implement Complex Math Equations in LaTeX. You can also use the link: https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols for...You've got a couple problems. (1) You're missing parentheses for the denominator of form2 (and R has no way to know that you want to add a constant a in the denominator, or where to put any of the parameters, really), and much more problematic: (2) your 2nd model isn't linear, so lm won't work.. Fixing (1) is easy: form2 <- h ~ 1.3 + …Question 10. In the x y -plane, the point ( p, r) lies on the line with equation y = x + b, where b is a constant. The point with coordinates ( 2 p, 5 r) lies on the line with equation y = 2 x + b. If p ≠ 0, what is the value of. r.The complex plane plays an important role in Mathematics. It is also known as z-plane, which is composed of two mutually perpendicular lines called axes.Solving quadratic equations can lead to complex solutions. Listen to Sal break down the process using the quadratic formula and standard form. Then, learn how to verify these …Launch the Bing app for iOS on your iPhone or iPad. Tap the camera icon and on the screen again select the setting for Math (between Auto and Barcode). Focus the camera on the math equation that may be on paper or on the whiteboard. The app can scan printed, typed, or handwritten equations. Bing detects the equation and calculates …Learn how to manipulate and solve equations involving complex numbers, which are numbers with real and imaginary parts. See examples of complex math …Select Insert > Equation or press Alt + =. To use a built-in formula, select Design > Equation. To create your own, select Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. Select Insert to bring your equation into the file. See the ribbon for more Structures and Convert options.7−i 2 +10i 7 − i 2 + 10 i Solution. 1 +5i −3i 1 + 5 i − 3 i Solution. 6 +7i 8−i 6 + 7 i 8 − i Solution. Here is a set of practice problems to accompany the Complex Numbers< section of the Preliminaries chapter of the notes for …The web page explains that the most complicated equation is the Standard Model Lagrangian, which describes the dynamics of every kind of particle and all …
The complex conjugate of a complex number is obtained by ... teacher of Mathematics. He never obtained the ... (a) The complex number z satisfies the equation z +1 .... Seattle sounders vs san jose
With Tenor, maker of GIF Keyboard, add popular Math Equation animated GIFs to your conversations. Share the best GIFs now >>> A simple formula is a mathematical expression with one operator, such as 7+9. A complex formula has more than one mathematical operator, such as 5+2*8. When there is more than one operation in a formula, the order of operations tells your spreadsheet which operation to calculate first. In order to use complex formulas, you will need to ... A constant in math is a fixed value. It may be a number on its own or a letter that stands for a fixed number in an equation. For example, in the equation “6x – 4 = 8,” both 4 and ...Select Insert > Equation or press Alt + =. To use a built-in formula, select Design > Equation. To create your own, select Insert > Equation > Insert New Equation. Use your finger, stylus, or mouse to write your equation. Select Insert to bring your equation into the file. See the ribbon for more Structures and Convert options.I want to write some complex mathematical equation in my web page. Is there any plugin or anything for this? html; equation; Share. edited Sep 4, 2023 at 9:03. ... Now you can write mathematical equations and formulae in your html or web page. Visit the above link to know more about how to write mathematical formulae using jqMath.Miners guess a random target number that solves an equation generated by the Bitcoin Protocol. Of course, computers make this guess, not people. ... Bitcoin miners are solving complex mathematical puzzles in a process called proof-of-work, which helps secure the Bitcoin network and create new bitcoins. These puzzles are based on …I want to write some complex mathematical equation in my web page. Is there any plugin or anything for this? html; equation; Share. edited Sep 4, 2023 at 9:03. ... Now you can write mathematical equations and formulae in your html or web page. Visit the above link to know more about how to write mathematical formulae using jqMath.There are a few ways to write Math equations in HTML: Use HTML Math symbols and tags to build simple equations. Refer to this list for the available Math symbols. For example, ∛ 27 = 3 will display as ∛ 27 = 3. For complicated equations, use a library to help – Check out MathJax. Lastly, use online generators to create images of equations ...Dec 28, 2023 · In 2019, mathematicians finally solved one of the hardest math problems —one that had stumped them for decades. It’s called a Diophantine Equation, and it’s sometimes known as the “summing ... 3,819 complex math formulas on white board stock photos, 3D objects, vectors, and illustrations are available royalty-free. Complex math formulas on whiteboard. Mathematics and science with economics concept. Real equations, symbols handwritten by a professional.In mathematics, Euler's identity [note 1] (also known as Euler's equation) is the equality. where. is Euler's number, the base of natural logarithms, is the imaginary unit, which by definition satisfies , and. is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler.With Tenor, maker of GIF Keyboard, add popular Math Equation animated GIFs to your conversations. Share the best GIFs now >>>Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the … Explanation: . To solve for , we must first solve the equation with the complex number for and .We therefore need to match up the real portion of the compex number with the real portions of the expression, and the imaginary portion of the complex number with the imaginary portion of the expression. Hardest Math Problem Solved Diophantine Equation Answers. What is the most complicated equation complex math seemingly unsolvable hardest problem ever 17 equations that changed world do formulas in always solving any of these tough problems. ← Solving Basic Trig Equations Worksheet Pdf Example Of Quadratic Equation That …The values on the x, y-plane are real numbers, so the complex-valued solutions of the equation cannot be seen on the x-axis. Your complex-valued answer is still a valid "zero" or "root" or "solution" for that quadratic equation, because, if you plug the answer value back into the quadratic equation, you'll get zero after you simplify.Complex Numbers. Complex numbers are numbers of the form a + ⅈb, where a and b are real and ⅈ is the imaginary unit. They arise in many areas of mathematics, including algebra, calculus, analysis and the study of special functions, and across a wide range of scientific and engineering disciplines. Oftentimes, they form connections between ...Algebra 1 16 units · 184 skills. Unit 1 Algebra foundations. Unit 2 Solving equations & inequalities. Unit 3 Working with units. Unit 4 Linear equations & graphs. Unit 5 Forms of linear equations. Unit 6 Systems of equations. Unit 7 …7−i 2 +10i 7 − i 2 + 10 i Solution. 1 +5i −3i 1 + 5 i − 3 i Solution. 6 +7i 8−i 6 + 7 i 8 − i Solution. Here is a set of practice problems to accompany the Complex Numbers< section of the Preliminaries chapter of the notes for …In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, ... or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent ....