Skill 10 of 12

Solve quadratics with the quadratic formula

Solve any quadratic equation with real coefficients using the quadratic formula, giving real or complex solutions in simplest form.

Common Core standards: A-REI.4bN-CN.7
Builds on: Use the discriminant to classify solutions
Add, subtract, and multiply complex numbers

Common mistakes

What students most often get wrong with this skill:

Sign error on $-b$
Dividing only part of the numerator by $2a$
Not writing the square root of a negative discriminant as $i$ times a real number

A worked example

From the lesson, with every step explained.

Example 1: rational solutions

Problem: Solve $x^2-3x-10=0$ .

Plan: The equation already equals $0$ , so identify $a,b,c$ and use the formula.

1) Identify the coefficients.

$a=1,\ b=-3,\ c=-10$

2) Write the formula.

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

3) Substitute. Since $b=-3$ , we get $-b=3$ .

x=\frac{3\pm\sqrt{(-3)^2-4(1)(-10)}}{2(1)}

4) Simplify under the root.

x=\frac{3\pm\sqrt{9+40}}{2}=\frac{3\pm\sqrt{49}}{2}

5) Take the root: $\sqrt{49}=7$ .

x=\frac{3\pm 7}{2}

6) Split into two answers.

$x=\frac{3+7}{2}=5$

$x=\frac{3-7}{2}=-2$

Answer: $x=5$ or $x=-2$ .

Check: Put $x=5$ in: $25-15-10=0$ . Correct.

Put $x=-2$ in: $4+6-10=0$ . Correct.

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