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How to Solve Quadratic Equations
Last Updated: February 10, 2023 Fact Checked
This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 9 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,392,719 times.
A quadratic equation is a polynomial equation in a single variable where the highest exponent of the variable is 2. [1] X Research source There are three main ways to solve quadratic equations: 1) to factor the quadratic equation if you can do so, 2) to use the quadratic formula, or 3) to complete the square. If you want to know how to master these three methods, just follow these steps.
Factoring the Equation
- Then, use the process of elimination to plug in the factors of 4 to find a combination that produces -11x when multiplied. You can either use a combination of 4 and 1, or 2 and 2, since both of those numbers multiply to get 4. Just remember that one of the terms should be negative, since the term is -4. [3] X Research source
- 3x = -1 ..... by subtracting
- 3x/3 = -1/3 ..... by dividing
- x = -1/3 ..... simplified
- x = 4 ..... by subtracting
- x = (-1/3, 4) ..... by making a set of possible, separate solutions, meaning x = -1/3, or x = 4 seem good.
- So, both solutions do "check" separately, and both are verified as working and correct for two different solutions.
Using the Quadratic Formula
- 4x 2 - 5x - 13 = x 2 -5
- 4x 2 - x 2 - 5x - 13 +5 = 0
- 3x 2 - 5x - 8 = 0
- {-b +/-√ (b 2 - 4ac)}/2
- {-(-5) +/-√ ((-5) 2 - 4(3)(-8))}/2(3) =
- {-(-5) +/-√ ((-5) 2 - (-96))}/2(3)
- {-(-5) +/-√ ((-5) 2 - (-96))}/2(3) =
- {5 +/-√(25 + 96)}/6
- {5 +/-√(121)}/6
- (5 + 11)/6 = 16/6
- (5-11)/6 = -6/6
- x = (-1, 8/3)
Completing the Square
- 2x 2 - 9 = 12x =
- In this equation, the a term is 2, the b term is -12, and the c term is -9.
- 2x 2 - 12x - 9 = 0
- 2x 2 - 12x = 9
- 2x 2 /2 - 12x/2 = 9/2 =
- x 2 - 6x = 9/2
- -6/2 = -3 =
- (-3) 2 = 9 =
- x 2 - 6x + 9 = 9/2 + 9
- x = 3 + 3(√6)/2
- x = 3 - 3(√6)/2)
Practice Problems and Answers
Expert Q&A
- If the number under the square root is not a perfect square, then the last few steps run a little differently. Here is an example: [14] X Research source Thanks Helpful 2 Not Helpful 0
- As you can see, the radical sign did not disappear completely. Therefore, the terms in the numerator cannot be combined (because they are not like terms). There is no purpose, then, to splitting up the plus-or-minus. Instead, we divide out any common factors --- but ONLY if the factor is common to both of the constants AND the radical's coefficient. Thanks Helpful 1 Not Helpful 0
- If the "b" is an even number, the formula is : {-(b/2) +/- √(b/2)-ac}/a. Thanks Helpful 2 Not Helpful 0
You Might Also Like
- ↑ https://www.mathsisfun.com/definitions/quadratic-equation.html
- ↑ http://www.mathsisfun.com/algebra/factoring-quadratics.html
- ↑ https://www.mathportal.org/algebra/solving-system-of-linear-equations/elimination-method.php
- ↑ https://www.cuemath.com/algebra/quadratic-equations/
- ↑ https://www.purplemath.com/modules/solvquad4.htm
- ↑ http://www.purplemath.com/modules/quadform.htm
- ↑ https://uniskills.library.curtin.edu.au/numeracy/algebra/quadratic-equations/
- ↑ http://www.mathsisfun.com/algebra/completing-square.html
- ↑ http://www.umsl.edu/~defreeseca/intalg/ch7extra/quadmeth.htm
About This Article
To solve quadratic equations, start by combining all of the like terms and moving them to one side of the equation. Then, factor the expression, and set each set of parentheses equal to 0 as separate equations. Finally, solve each equation separately to find the 2 possible values for x. To learn how to solve quadratic equations using the quadratic formula, scroll down! Did this summary help you? Yes No
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A quadratic equation is an equation that could be written as
ax 2 + bx + c = 0
There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.
To solve a quadratic equation by factoring,
- Put all terms on one side of the equal sign, leaving zero on the other side.
- Set each factor equal to zero.
- Solve each of these equations.
- Check by inserting your answer in the original equation.
Solve x 2 – 6 x = 16.
Following the steps,
x 2 – 6 x = 16 becomes x 2 – 6 x – 16 = 0
( x – 8)( x + 2) = 0
Both values, 8 and –2, are solutions to the original equation.
Solve y 2 = – 6 y – 5.
Setting all terms equal to zero,
y 2 + 6 y + 5 = 0
( y + 5)( y + 1) = 0
To check, y 2 = –6 y – 5
A quadratic with a term missing is called an incomplete quadratic (as long as the ax 2 term isn't missing).
Solve x 2 – 16 = 0.
To check, x 2 – 16 = 0
Solve x 2 + 6 x = 0.
To check, x 2 + 6 x = 0
Solve 2 x 2 + 2 x – 1 = x 2 + 6 x – 5.
First, simplify by putting all terms on one side and combining like terms.
Now, factor.
To check, 2 x 2 + 2 x – 1 = x 2 + 6 x – 5
The quadratic formula
a, b, and c are taken from the quadratic equation written in its general form of
where a is the numeral that goes in front of x 2 , b is the numeral that goes in front of x , and c is the numeral with no variable next to it (a.k.a., “the constant”).
When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b 2 – 4 ac . A quadratic equation with real numbers as coefficients can have the following:
- Two different real roots if the discriminant b 2 – 4 ac is a positive number.
- One real root if the discriminant b 2 – 4 ac is equal to 0.
- No real root if the discriminant b 2 – 4 ac is a negative number.
Solve for x : x 2 – 5 x = –6.
Setting all terms equal to 0,
x 2 – 5 x + 6 = 0
Then substitute 1 (which is understood to be in front of the x 2 ), –5, and 6 for a , b , and c, respectively, in the quadratic formula and simplify.
Because the discriminant b 2 – 4 ac is positive, you get two different real roots.
Example produces rational roots. In Example , the quadratic formula is used to solve an equation whose roots are not rational.
Solve for y : y 2 = –2y + 2.
y 2 + 2 y – 2 = 0
Then substitute 1, 2, and –2 for a , b , and c, respectively, in the quadratic formula and simplify.
Note that the two roots are irrational.
Solve for x : x 2 + 2 x + 1 = 0.
Substituting in the quadratic formula,
Since the discriminant b 2 – 4 ac is 0, the equation has one root.
The quadratic formula can also be used to solve quadratic equations whose roots are imaginary numbers, that is, they have no solution in the real number system.
Solve for x : x ( x + 2) + 2 = 0, or x 2 + 2 x + 2 = 0.
Since the discriminant b 2 – 4 ac is negative, this equation has no solution in the real number system.
Completing the square
A third method of solving quadratic equations that works with both real and imaginary roots is called completing the square.
- Put the equation into the form ax 2 + bx = – c .
- Find the square root of both sides of the equation.
- Solve the resulting equation.
Solve for x : x 2 – 6 x + 5 = 0.
Arrange in the form of
Take the square root of both sides.
x – 3 = ±2
Solve for y : y 2 + 2 y – 4 = 0.
Solve for x : 2 x 2 + 3 x + 2 = 0.
There is no solution in the real number system. It may interest you to know that the completing the square process for solving quadratic equations was used on the equation ax 2 + bx + c = 0 to derive the quadratic formula.
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Quadratics or Quadratic Equations
Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations . The general form of the quadratic equation is:
ax² + bx + c = 0
where x is an unknown variable and a, b, c are numerical coefficients. For example, x 2 + 2x +1 is a quadratic or quadratic equation. Here, a ≠ 0 because if it equals zero then the equation will not remain quadratic anymore and it will become a linear equation, such as:
Thus, this equation cannot be called a quadratic equation.
The terms a, b and c are also called quadratic coefficients.
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations . The roots of any polynomial are the solutions for the given equation.
What is Quadratic Equation?
The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:
where x is the unknown variable and a, b and c are the constant terms.
Standard Form of Quadratic Equation
Since the quadratic includes only one unknown term or variable, thus it is called univariate. The power of variable x is always non-negative integers. Hence the equation is a polynomial equation with the highest power as 2.
The solution for this equation is the values of x, which are also called zeros. Zeros of the polynomial are the solution for which the equation is satisfied. In the case of quadratics, there are two roots or zeros of the equation. And if we put the values of roots or x on the left-hand side of the equation, it will equal to zero. Therefore, they are called zeros.
Quadratics Formula
The formula for a quadratic equation is used to find the roots of the equation. Since quadratics have a degree equal to two, therefore there will be two solutions for the equation. Suppose ax² + bx + c = 0 is the quadratic equation, then the formula to find the roots of this equation will be:
x = [-b±√(b 2 -4ac)]/2a
The sign of plus/minus indicates there will be two solutions for x. Learn in detail the quadratic formula here.
Examples of Quadratics
Beneath are the illustrations of quadratic equations of the form (ax² + bx + c = 0)
- x² –x – 9 = 0
- 5x² – 2x – 6 = 0
- 3x² + 4x + 8 = 0
- -x² +6x + 12 = 0
Examples of a quadratic equation with the absence of a ‘ C ‘- a constant term.
- -x² – 9x = 0
- x² + 2x = 0
- -6x² – 3x = 0
- -5x² + x = 0
- -12x² + 13x = 0
- 11x² – 27x = 0
Following are the examples of a quadratic equation in factored form
- (x – 6)(x + 1) = 0 [ result obtained after solving is x² – 5x – 6 = 0]
- –3(x – 4)(2x + 3) = 0 [result obtained after solving is -6x² + 15x + 36 = 0]
- (x − 5)(x + 3) = 0 [result obtained after solving is x² − 2x − 15 = 0]
- (x – 5)(x + 2) = 0 [ result obtained after solving is x² – 3x – 10 = 0]
- (x – 4)(x + 2) = 0 [result obtained after solving is x² – 2x – 8 = 0]
- (2x+3)(3x – 2) = 0 [result obtained after solving is 6x² + 5x – 6]
Below are the examples of a quadratic equation with an absence of linear co – efficient ‘ bx’
- 2x² – 64 = 0
- x² – 16 = 0
- 9x² + 49 = 0
- -2x² – 4 = 0
- 4x² + 81 = 0
- -x² – 9 = 0
How to Solve Quadratic Equations?
There are basically four methods of solving quadratic equations. They are:
- Completing the square
Using Quadratic Formula
- Taking the square root
Factoring of Quadratics
- Begin with a equation of the form ax² + bx + c = 0
- Ensure that it is set to adequate zero.
- Factor the left-hand side of the equation by assuming zero on the right-hand side of the equation.
- Assign each factor equal to zero.
- Now solve the equation in order to determine the values of x.
Suppose if the main coefficient is not equal to one then deliberately, you have to follow a methodology in the arrangement of the factors.
(2x+3)(x-2)=0
Learn more about the factorization of quadratic equations here.
Completing the Square Method
Let us learn this method with example.
Example: Solve 2x 2 – x – 1 = 0.
First, move the constant term to the other side of the equation.
2x 2 – x = 1
Dividing both sides by 2.
x 2 – x/2 = ½
Add the square of half of the coefficient of x, (b/2a) 2 , on both the sides, i.e., 1/16
x 2 – x/2 + 1/16 = ½ + 1/16
Now we can factor the right side,
(x-¼) 2 = 9/16 = (¾) 2
Taking root on both sides;
X – ¼ = ±3/4
Add ¼ on both sides
X = ¼ + ¾ = 4/4 = 1
X = ¼ – ¾ = -2/4 = -½
To learn more about completing the square method, click here .
For the given Quadratic equation of the form, ax² + bx + c = 0
Therefore the roots of the given equation can be found by:
\(\begin{array}{l}x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\end{array} \)
where ± (one plus and one minus) represent two distinct roots of the given equation.
Taking the Square Root
We can use this method for the equations such as:
x 2 + a 2 = 0
Example: Solve x 2 – 50 = 0.
x 2 – 50 = 0
Taking the roots both sides
√x 2 = ±√50
x = ±√(2 x 5 x 5)
Thus, we got the required solution.
Related Articles
Video lesson on quadratic equations, range of quadratic equations.
Solved Problems on Quadratic Equations
Applications of quadratic equations.
Many real-life word problems can be solved using quadratic equations. While solving word problems, some common quadratic equation applications include speed problems and Geometry area problems.
- Solving the problems related to finding the area of quadrilateral such as rectangle, parallelogram and so on
- Solving Word Problems involving Distance, speed, and time, etc.,
Example: Find the width of a rectangle of area 336 cm2 if its length is equal to the 4 more than twice its width. Solution: Let x cm be the width of the rectangle. Length = (2x + 4) cm We know that Area of rectangle = Length x Width x(2x + 4) = 336 2x 2 + 4x – 336 = 0 x 2 + 2x – 168 = 0 x 2 + 14x – 12x – 168 = 0 x(x + 14) – 12(x + 14) = 0 (x + 14)(x – 12) = 0 x = -14, x = 12 Measurement cannot be negative. Therefore, Width of the rectangle = x = 12 cm
Practice Questions
- Solve x 2 + 2 x + 1 = 0.
- Solve 5x 2 + 6x + 1 = 0
- Solve 2x 2 + 3 x + 2 = 0.
- Solve x 2 − 4x + 6.25 = 0
Frequently Asked Questions on Quadratics
What is a quadratic equation, what are the methods to solve a quadratic equation, is x 2 – 1 a quadratic equation, what is the solution of x 2 + 4 = 0, write the quadratic equation in the form of sum and product of roots., leave a comment cancel reply.
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x=√9 Squaring both the sides, x^2 = 9 x^2 – 9 = 0 It is a quadratic equation.
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Solving quadratics by factoring Google Classroom About Transcript Sal solves the equation s^2-2s-35=0 by factoring the expression on the left as (s+5) (s-7) and finding the s-values that make each factor equal to zero. Created by Sal Khan and Monterey Institute for Technology and Education. Questions Tips & Thanks Want to join the conversation?
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1 concept Introduction to Quadratic Equations 5m 1 Comment Mark as completed Was this helpful? 13 2 Problem Write the given quadratic equation in standard form. Identify a, b, and c. −4x2 +x = 8 A a = - 4, b = 0, c = - 8 B a = - 4, b = 1, c = 8 C a = - 4, b = 1, c = - 8 D a = 2, b = 1, c = 0 0 Comments
A quadratic equation is an equation that could be written as ax 2 + bx + c = 0 when a 0. There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square. Factoring. To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other ...
There are many ways to solve quadratics. All quadratic equations can be written in the form \ (ax^2 + bx + c = 0\) where \ (a\), \ (b\) and \ (c\) are numbers (\ (a\) cannot be equal to 0, but...
Quadratics Formula. The formula for a quadratic equation is used to find the roots of the equation. Since quadratics have a degree equal to two, therefore there will be two solutions for the equation. Suppose ax² + bx + c = 0 is the quadratic equation, then the formula to find the roots of this equation will be: x = [-b±√ (b2-4ac)]/2a.
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11 years ago The point of the zero-product property is this: If two or more factors are multiplied together to make 0, then one of the factors must = 0. Think about it, if you want to make 0 by multiplying, you have to have a 0 as a factor: 0 * 8 = 0 125 * 0 = 0 1/4 * 0 = 0 2 * 3 * 4 * 5 * 6 * 0 = 0
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