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Using the Quadratic Equation Formula
by Ron Kurtus (revised 28 December 2010)
Sometimes a quadratic equation (ax2 + bx + c = 0) can be solved for x by factoring. But often factoring the quadratic expression is not easy or possible. In such cases, the quadratic formula is used to solve the equation. That formula is x = [−b ± √(b2 − 4ac)]/2a. Besides having solutions consisting of rational numbers, solutions of quadratic equations can be irrational or even imaginary.
Questions you may have include:
- How do you solve equations using the quadratic formula?
- What are irrational solutions?
- What are imaginary solutions?
This lesson will answer those questions.
Solving quadratic equations
You can solve the quadratic equation ax2 + bx + c = 0 that is in the proper format where a, b and c are whole numbers and a > 0, by using the quadratic formula:
x = [−b ± √(b2 − 4ac)]/2a
It is good to memorize the equation in words:
"x equals minus b plus-or-minus the square root of b-squared minus 4ac, divided by 2a."
Rational solutions
Often the solutions to quadratic equations are rational numbers, which are integers or fractions. The requirement for the solution to be an integer or fraction is that √(b2 − 4ac) is a whole number.
An example is the solution to the equation x2 + 2x − 15, which is:
x = [−2 ± √(22 − 4*{−15})]/2
x = [−2 ± √(4 + 60)]/2
x = [−2 ± √(64)]/2
x = [−2 ± 8]/2
x = −10/2 and x = +6/2
x = −5 and x = 3
Try the equation 2x2 −x − 1:
x = [−1 ± √(12 − 4*2*{−1})]/4
x = [−1 ± √(1 + 8)]/4
x = [−1 ± √(9)]/4
x = [−1 ± 3)]/4
x = −4/4 and x = 2/4
x = −1 and x = 1/2
Irrational solution
The solution to some quadratic equations consist of irrational values for x. In other words, the square root of b2 − 4ac is not a whole number. For example, √2 is an irrational number equal to 1.41412....
Consider the equation x2 + 3x + 1 = 0:
x = [−3 ± √(32 − 4)]/2
x = [−3 ± √(9 − 4)]/2
x = [−3 ± √5]/2
x = −3/2 + (√5)/2 and x = −3/2 − (√5)/2
Both solutions are irrational numbers.
Imaginary solution
An imaginary number is a multiple of √−1. It is called imaginary, since no number exists whose square is −1. Imaginary numbers are used in certain equations in electrical engineering, signal processing and quantum mechanics.
Consider the equation x2 + x + 1 = 0:
x = [−1 ± √(12 − 4)]/2
x = [−1 ± √−3]/2
x = −1/2 + (√−3)/2 and x = −1/2 − (√−3)/2
Both solutions are imaginary numbers.
Summary
The quadratic formula is used when the solution to a quadratic equation cannot be readily solved by factoring. It is worthwhile to memorize the quadratic formula. Besides having solutions consisting of rational numbers, solutions of quadratic equations can be irrational or even imaginary.
Use tools that make your work easier
Resources and references
The following resources provide information on this subject:
Websites
Quadratic Equations - From AlgebraHelp.com
Quadratic Equation Calculator - From Math.com
Books
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Using the Quadratic Equation Formula