Algebra II

The Quadratic Formula

Lesson

The quadratic formulasolves any quadratic equation — even ones that don’t factor nicely. Given:

ax^2 + bx + c = 0

the solutions are:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The $\pm$ means there are two solutions: one with + on top of the radical, one with −. The expression under the radical $(b^2 - 4ac)$ is called the discriminant.

To use the formula:

Make sure the equation is in the form $ax^2 + bx + c = 0$ (everything on one side, zero on the other).
Identify $a$ , $b$ , and $c$ , including signs.
Plug into the formula and simplify.

Worked example 1

x^2 + 5x + 6 = 0

Identify: $a = 1, b = 5, c = 6$ .

x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2}

Two solutions:

x = \frac{-5 + 1}{2} = -2 \quad \text{or} \quad x = \frac{-5 - 1}{2} = -3

Worked example 2

2x^2 - 7x + 3 = 0

$a = 2, b = -7, c = 3$ .

x = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm 5}{4}

x = 3 \quad \text{or} \quad x = \frac{1}{2}

How to type your answer

Enter both solutions separated by a comma— order doesn’t matter. Use fractions where helpful (e.g. 1/2). Examples: -2,-3, 3,1/2, 5,-1.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x^2 + 5x + 6 = 0

Problem 2

x^2 - 5x + 6 = 0

Problem 3

x^2 + x - 12 = 0

Problem 4

x^2 - 9 = 0

Practice

Standard problems matching the lesson.

Problem 5

x^2 + 7x + 10 = 0

Problem 6

x^2 - 8x + 15 = 0

Problem 7

x^2 - 4x - 21 = 0

Problem 8

x^2 + 6x + 5 = 0

Problem 9

2x^2 - 7x + 3 = 0

Problem 10

2x^2 + 5x - 3 = 0

Problem 11

3x^2 - 7x + 2 = 0

Problem 12

x^2 + 2x - 15 = 0

Problem 13

x^2 - 10x + 25 = 0

Problem 14

x^2 + 4x + 4 = 0

Problem 15

2x^2 - 5x - 3 = 0

Problem 16

3x^2 + 4x + 1 = 0

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x^2 = 5x + 14

Problem 18

x^2 + 9x = -20

Problem 19

4x^2 - 4x + 1 = 0

Problem 20

6x^2 + x - 1 = 0

Problem 21

x^2 - 6x + 8 = 0

Problem 22

5x^2 - 14x - 3 = 0