The Quadratic Formula and the Discriminant

One formula, and a number that predicts the answer.

12 problems · about 22 minutes · A-REI.B.4, N-CN.C.7

What this lesson teaches

The student solves quadratic equations with the formula and uses the discriminant to predict the number and type of roots.

The quadratic formula solves every quadratic, factorable or not.
The discriminant b² − 4ac predicts two real, one repeated, or two complex roots.
Complex roots always arrive as conjugate pairs.

Straightforward practice. Get the method working first.

5 problems

x² + 3x + 2 = 0. What is the discriminant?
Answer 1
Why 1, so two rational roots.
A negative discriminant means what about the graph?
Answer The parabola never crosses the x-axis.
Why It misses the x-axis entirely.
x² − 6x + 5 = 0. Discriminant?
Answer 16
Why 36 − 20.
x² − 6x + 5 = 0. The larger root?
Answer 5
Why (6 + 4) ÷ 2.
x² + 2x + 10 = 0. Discriminant?
Answer -36
Why 4 − 40.

The same ideas with more to keep track of.

3 problems

x² − 10x + 25 = 0. How many distinct real roots?
Answer 1
Why The discriminant is zero.
2x² − 4x − 6 = 0. The larger root?
Answer 3
Why Divide through by 2 first: x² − 2x − 3 = 0.
x² + 9 = 0. One root is 3i. What is the other, as a coefficient of i?
Answer -3
Why Complex roots arrive as conjugate pairs.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each discriminant by what it predicts.
Answer Two real roots: b² − 4ac = 25, b² − 4ac = 3 · One repeated root: b² − 4ac = 0 · Two complex roots: b² − 4ac = −9
Why Only the sign matters, not the size.
x² − 2x + 5 = 0. One root is 1 + 2i. What is the real part of the other root?
Answer 1
Why The conjugate keeps the real part.
The Predictor: x² + 4x + 4 = 0. What is the discriminant?
Answer 0
Why 0, so there is one repeated root.
The Ball: A ball follows h = −5t² + 20t. At what time, in seconds, does it return to the ground, other than t = 0?
Answer 4 seconds
Why 4 seconds.