Bluebook SAT Test Nonlinear Equations and Systems 1

Question 1

What are the solutions to x² − 5x + 6 = 0?

Accepted Answer

x = 2 and x = 3

Answer

Factor as (x−2)(x−3) = 0, giving x = 2 and x = 3.

Question 2

The quadratic equation x² − 4x + k = 0 has exactly one real solution. What is the value of k?

Accepted Answer

4

Answer

For exactly one solution the discriminant = 0: (−4)² − 4(1)(k) = 0 → 16 = 4k → k = 4.

Question 3

Which of the following are both solutions to 2x² + 3x − 2 = 0?

Accepted Answer

x = 1/2 and x = −2

Answer

Factor: (2x−1)(x+2) = 0 → x = 1/2 or x = −2.

Question 4

How many real solutions does the equation x² + 6x + 10 = 0 have?

Accepted Answer

None

Answer

Discriminant = 6² − 4(1)(10) = 36 − 40 = −4 < 0, so no real solutions exist.

Question 5

What is the vertex of the parabola y = x² − 8x + 15?

Accepted Answer

(4, −1)

Answer

x-vertex = −(−8)/(2·1) = 4; y = 16 − 32 + 15 = −1; vertex = (4, −1).

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