Quadratic Formula: Equation & Examples

Let’s learn how to use the quadratic formula.

Using the quadratic formula is one of the methods used in solving quadratic equations, especially when the equation cannot be solved by factoring.

The quadratic formula solves any quadratic equation in the standard form: ax²+bx+c = 0, where a ≠ 0. So, make sure the equation you’re working on is in the standard form.

As shown in the photo above, the quadratic formula is:

x = −b ± √(b²− 4ac) / 2a

Where:
a = the coefficient in front of or the number beside x²
b = the coefficient in front of or the number beside x
c = the constant

How to Solve an Equation using the Quadratic Formula:

First, identify the values of a, b, and c in the given equation then substitute them to the quadratic formula. Lastly, perform the needed operations.

But before solving the equation, remember:

1. Your equation must be arranged in the standard quadratic form: ax²+bx+c=0.

2. The 2a in the denominator of the quadratic formula is underneath everything above.

3. Make sure not to drop the square root or the plus (+) and minus (-) signs in the middle of your calculations.

Example #1:

Look for the value of x in the equation, x²+9x+14 = 0

Step 1: Identify the values of a, b, & c.
a = 1, b = 9, c = 14

Step 2: Write the quadratic formula.
x = −b ± √(b²− 4ac) / 2a

Step 3: Substitute all the values in the formula.
x = −9 ± √(9²− 4(1)(14)) / 2(1)

Step 4: Simplify the fraction.
x = -9 ± √(81− 56) / 2
x = −9± √25 / 2
Remember to solve using both addition and subtraction.
x = -9+5 / 2 or x = -9-5 / 2
x = -4 / 2 or x = -14 / 2

Step 5: Solve for x.
x = -2 and x = – 7

Therefore, the roots of the equation are -2 and -7.

Example #2:

Look for the value of x in the equation, 2x² –x – 1 = 0

Step 1: Identify the values of a, b, & c.
a = 2, b = -1, c = -1

Step 2: Write the quadratic formula.
x = −b ± √(b²− 4ac) / 2a

Step 3: Substitute all the values in the formula.
x = −(-1) ± √(-1²− 4(2)(-1)) / 2(2)

Step 4: Simplify the fraction.
x = 1 ± √1+8 / 4
x = 1 ± √9 / 4
Remember to solve using both addition and subtraction.
x = 1+3 / 4 or x = 1-3 / 4
x = 4 / 4 or x = -2 / 4

Step 5: Solve for x.
x = 1 and x = – 1/2

Therefore, the roots of the equation are 1 and – 1/2 .

Related Reading: Interval Notation – Writing & Graphing

Thank you for reading. We hope it’s effective! Always feel free to revisit this page if you ever have any questions about the quadratic formula.