Explanation of Quadratic Equations - Succeed in Understanding Algebra. Also refer to one variable, second degree, solution, expression, quadratic formula, format, mathematics, math, maths, Ron Kurtus, School for Champions. Copyright © Restrictions
Quadratic Equations
by Ron Kurtus (14 September 2008)
A quadratic equation is an 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 with a not equal to 0.
Sometimes the expressions in an equation with one variable that contains an x2 element must be rearranged to get it in the standard quadratic equation format.
You can solve this type of equation by using factoring or the quadratic formula. The equation has two solutions or values for x.
Questions you may have include:
- What do the items in the quadratic equation mean?
- What is the proper format for a quadratic equation?
- How can the equation be solved?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Equation
The quadratic equation is in the form of ax2 + bx + c = 0, where:
- a, b and c represent positive or negative numbers
- a is not equal to 0, written as a ≠ 0
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 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)
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.)
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:
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 subtracting7 + 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.
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.
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 factored into
(x + 1)(x + 2) = 0.
This leads to the solution of x = −1 and x = −2.
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 or the quadratic formula to solve a quadratic equation.
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Use all your knowledge
Resources
The following resources provide information on this subject:
Websites
Books
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.
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Quadratic Equations