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Quadratic Equations

by Ron Kurtus (revised 7 December 2014)

A quadratic equation is an Algebraic equation with one variable that can be put in the form of
ax2 + bx + c = 0, where x is the variable and a, b and c are constants, and a is not equal to 0.

Sometimes the expressions in a quadratic equation are in a different order and should be rearranged to get it in the standard quadratic equation format.

You can solve this type of equation by using factoring, least squares method, or the quadratic formula. The equation has two solutions or values for x.

Questions you may have include:

This lesson will answer those questions.



Form of quadratic equation

The quadratic equation is in the form of:

ax2 + bx + c = 0

where

A quadratic equation is considered an equation of the second degree, because of the x2 term. If a = 0, the equation becomes an equation of the first degree or a linear equation of the form bx + c = 0.

Examples of quadratic equations

Examples of quadratic equations include:

x2 + x + 1 = 0 (where a = 1, b = 1, and c = 1)

3x2 − 6x + 5 = 0 (where a = 3, b = −6, and c = 5)

2x2 + 3x = 0 (where a = 2, b = 3, and c = 0)

25x2 − 64 = 0 (where a = 25, b = 0, and c = − 64)

Equations that are not quadratic equations

The following equations are not quadratic equations, because a = 0.

2x + 7 = 0

3x2 − 3x2 + x + 5 = 0 because 3x2 − 3x2 = 0, resulting in x + 5 = 0

These equations are linear equations. (See Linear Equations for more information.)

Put in proper format

There is a definite format for the quadratic equation.

Equation must equal 0

The expression on the right side of the equal sign must equal 0 to be in the proper quadratic equation format. In other words:

The equation 3x2 + x + 1 = x2 − 4 should be put in the form of x2 + x + 5 = 0 by subtracting x2 − 4 from both sides of the equation.

2x2 − x = 7 + 3x should be put in the form of 2x2 − 4x − 7 = 0 by subtracting
7 + 3x from both sides of the equation.

"a" should be positive

Although a can be either positive or negative, it is preferred to put the equation in the form where a is a positive number.

If a is negative, you can multiply both sides of the equation by −1.

For example, you can multiply both sides of the equation
−2x2 + x − 4 = 0
by −1 so that it becomes 2x2 − x + 4 = 0.

Usually integers

Usually, a, b and c are integers. If they are fractions or decimals, it is desirable to multiply the equation by some number to make a, b and c integers.

For example, consider the equation 0.1x2 + 5.3x − 0.4 = 0. You should multiply each side of the equation by 10 to put it in the more desirable form of x2 + 53x − 4 = 0.

Likewise, x2 + x/3 + 1/4 = 0, should be multiplied by 12 to put it in the form of
12x2 + 4x + 3 = 0
.

Finding solutions

A major objective in Algebra is to find the solutions to equations. In other words, for a quadratic equation, you want to find the values of x that would result in ax2 + bx + c equaling 0.

For example, solutions to the equation x2 + 3x + 2 = 0 are x = −1 and x = −2.

You can substitute each number back into the equation to verify that they are solutions.

Note that for a quadratic equation, there are usually two solutions. In some cases, the two solutions are the same, so it is actually just one solution.

You can use trial and error to find solutions to a quadratic equation, but that certainly is not the best method to use. Standard methods to solve quadratic equations are:

Factoring

One common method to solve a quadratic equation is by factoring the left expression into two sub-expressions and then solving each of those.

For example, x2 + 3x + 2 = 0 can be readily factored into (x + 1)(x + 2) = 0.

This leads to the solutions of x = −1 and x = −2.

Completing the square

Completing the square is another way to find solutions. The method involves rearranging the equation and adding a term to both sides of the equal sign in order to make the left side a squared expression. Then by taking the square root, you can get your solutions.

(See Solving Quadratic Equations by Completing the Square Method for more information.)

Quadratic formula

Another method is to use the quadratic formula, which states that for the quadratic equation ax2 + bx + c = 0, the solutions for x can be determined from the formula:

x = [−b ±√(b2 −4ac)]/2a

(See Using the Quadratic Equation Formula for more information.)

Summary

A quadratic equation is in the form of ax2 + bx + c = 0. There are preferred ways a quadratic equation should be formatted. The goal is to find values of x that will provide a solution to the equation. You can use trial and error, factoring, completing the square, or the quadratic formula to solve a quadratic equation.


Use all your knowledge


Resources and references

Ron Kurtus' Credentials

Websites

Quadratic Equations - The MathPage/com

Quadratic Equations - MathIsFun.com

Quadratic equation - Wikipedia

Algebra Resources

Books

Top-rated books on Algebra


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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