Quadratic Functions and Equations in One Variable

Question 1

Solve x² − 5x + 6 = 0. What are the roots?

Accepted Answer

x = 2 or x = 3 — (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Question 2

Solve x² + 2x − 8 = 0. What are the roots?

Accepted Answer

x = −4 or x = 2 — (x + 4)(x − 2) = 0, so x = −4 or x = 2.

Question 3

For a quadratic function f(x) = ax² + bx + c, the graph opens downward when:

Accepted Answer

a < 0 — When a < 0 the parabola opens downward and has a maximum value.

Question 4

The value b² − 4ac is called the:

Accepted Answer

Discriminant — b² − 4ac is the discriminant, which tells the nature of the roots.

Question 5

How many real roots does x² − 4x + 4 = 0 have?

Accepted Answer

One repeated (equal) root — b² − 4ac = 16 − 16 = 0, so there are two equal roots (x = 2).

Question 6

For which value of the discriminant does a quadratic equation have NO real roots?

Accepted Answer

b² − 4ac < 0 — When the discriminant is negative there are no real roots.

Question 7

Solve x² − 9 = 0.

Accepted Answer

x = ±3 — x² = 9, so x = 3 or x = −3.

Question 8

Where does the graph of f(x) = x² + bx + c cut the y-axis?

Accepted Answer

At y = c — At x = 0, f(0) = c, so the y-intercept is c.

Question 9

Solve x² − 4x = 0.

Accepted Answer

x = 0 or x = 4 — x(x − 4) = 0, so x = 0 or x = 4.

Question 10

Using the quadratic formula for 2x² + 3x − 2 = 0, the roots are:

Accepted Answer

x = ½ or x = −2 — (2x − 1)(x + 2) = 0, so x = ½ or x = −2.