6.3: The Discriminant | Algebra 1 Honors

How it works

The discriminant is the expression under the square root in the quadratic formula. Its sign determines the number of real roots. A positive nonsquare value gives two irrational real roots; zero gives one repeated real root; a negative value gives no real roots.

D=b^2-4ac

Worked examples

Example 1

Classify the real roots of $\displaystyle x^2-4x+4=0$ .

$\displaystyle D=16-16=0$ , so there is one repeated real root, $\displaystyle x=2$ .

Example 2

Classify the real roots of $\displaystyle 2x^2+x+3=0$ .

$\displaystyle D=1-24=-23<0$ , so there are no real roots.

Problems

Problem 6.3.1

How many real roots does $\displaystyle x^2-3x-1=0$ have?

Show worked solution

$\displaystyle D=9+4=13>0$ , so there are two distinct real roots.

Problem 6.3.2

Find $\displaystyle k$ so that $\displaystyle x^2+6x+k=0$ has a repeated root.

Show worked solution

A repeated root requires $\displaystyle D=36-4k=0$ . Thus $\displaystyle k=9$ .

Problem 6.3.3

Use the graph of $\displaystyle y=x^2+2$ to check the discriminant prediction for its zeros.

The upward parabola y = x squared + 2 stays above the horizontal axis. — Compare the graph with the sign of the discriminant.

Show worked solution

Here $\displaystyle D=0^2-4(1)(2)=-8<0$ . The graph stays above the x-axis, so there are no real zeros.

The upward parabola y = x squared + 2 stays above the horizontal axis. The discriminant is −8 and there are no real zeros. — The discriminant is −8 and there are no real zeros.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 6.3.1

Problem 6.3.2

Problem 6.3.3

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