Bluebook SAT Test Nonlinear Equations and Systems 16

Question 1

If y = x² + 3 and y = −x² + 11, at which x-values do the two parabolas intersect?

Accepted Answer

x = ±2

Answer

x² + 3 = −x² + 11 → 2x² = 8 → x² = 4 → x = ±2.

Question 2

Which equation has a parabola that opens downward with vertex at (1, 4)?

Accepted Answer

y = −(x − 1)² + 4

Answer

Vertex form y = a(x − h)² + k with vertex (1, 4) and a < 0 (opens down): y = −(x − 1)² + 4.

Question 3

The parabola y = x² + 6x + 9 is tangent to the x-axis. At what x-value does this occur?

Accepted Answer

x = −3

Answer

x² + 6x + 9 = (x + 3)²; this equals 0 only at x = −3, confirming tangency at a single point.

Question 4

The sum of the squares of two consecutive integers is 85. What are the integers?

Accepted Answer

6 and 7

Answer

n² + (n+1)² = 85 → 2n² + 2n + 1 = 85 → 2n² + 2n − 84 = 0 → n² + n − 42 = 0 → (n+7)(n−6) = 0; n = 6, so integers are 6 and 7.

Question 5

For what value of c does x² + 8x + c = 0 have a double root?

Accepted Answer

16

Answer

Discriminant = 0: 64 − 4c = 0 → c = 16.

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