Skill 9 of 12

Use the discriminant to classify solutions

Compute $b^2-4ac$ and use it to determine whether a quadratic equation has two real, one real, or two complex (non-real) solutions.

Common Core standards: A-REI.4b
Builds on: The imaginary unit and square roots of negative numbers

Common mistakes

What students most often get wrong with this skill:

Computing $b^2$ as $-b^2$ when $b$ is negative
Using the discriminant before writing the equation in standard form
Believing a negative discriminant means there are no solutions at all

A worked example

From the lesson, with every step explained.

Example 1: compute D

Problem: Find the discriminant of $x^2-4x+1=0$ and say what kind of solutions it has.

Plan: The equation is already in standard form, so read off $a$ , $b$ , $c$ , then use $D=b^2-4ac$ .

1) Read the coefficients: $a=1$ , $b=-4$ , $c=1$ .
2) Write the formula with parentheses around each number:

D=(-4)^2-4(1)(1)

3) Square first. Since $(-4)(-4)=16$ :

D=16-4

4) Subtract:

D=12

Answer: $D=12>0$ , so there are two real solutions.

Check: The quadratic formula gives $x=\frac{4\pm\sqrt{12}}{2}=2\pm\sqrt{3}$ . These are two different real numbers, which matches $D>0$ .

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