Algebra · Quick Reference

Quadratic Formula Cheat Sheet

Algebra I · Lesson 9/17

In one line: the quadratic formula solves ANY quadratic equation, even ones that can't be factored nicely — a universal tool that always works as long as you know a, b, and c.

Key Ideas

1The Quadratic Formula. For ax^2+bx+c=0, x = [-b plus or minus sqrt(b^2-4ac)] / (2a).

2Identifying a, b, c. Match your equation to standard form ax^2+bx+c=0 and read off the three coefficients directly.

3The Discriminant. The expression b^2-4ac inside the square root. It determines the nature of the solutions without fully solving the equation.

4Interpreting the Discriminant. If b^2-4ac > 0: two distinct real solutions. If = 0: exactly one repeated real solution. If < 0: no real solutions (two complex solutions).

5When to Use the Quadratic Formula. Use it whenever factoring is difficult or impossible — it always works for any quadratic equation.

Worked Examples

Solve x^2 + 5x + 6 = 0 using the quadratic formula.

x = -2 or x = -3

Find the discriminant of 2x^2 + 3x + 5 = 0 and describe the solutions.

Discriminant = -31 (negative), so there are no real solutions

Solve x^2 - 4x + 4 = 0 using the quadratic formula.

x = 2 (a repeated root, since discriminant = 0)

Formulas

x = [-b plus or minus sqrt(b^2 - 4ac)] / (2a)

Discriminant = b^2 - 4ac

Practice Yourself

Find a, b, c for 3x^2 - 2x + 7 = 0.

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

Find the discriminant of x^2 + 2x + 1 = 0.

Find the discriminant of x^2 + x + 1 = 0.

-3