Nonlinear Equations and Systems Flashcards

A rocket's height in feet is h(t) = −16t² + 80t. What is the maximum height the rocket reaches?

Answer: 100 feet

Vertex time: t = −80/(2 × −16) = 2.5 s; h(2.5) = −16(6.25) + 80(2.5) = −100 + 200 = 100 feet.

How many solutions does the system y = x² − 2x + 1 and y = 4 have?

Answer: 2 solutions

x² − 2x + 1 = 4 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0; two solutions: x = 3 and x = −1.

The equation x² − kx + 9 = 0 has two equal real roots. What is the positive value of k?

Answer: 6

For equal roots, discriminant = 0: k² − 36 = 0; k² = 36; k = 6.

Which value of x satisfies both x² = 2x + 8 and x > 0?

Answer: x = 4

x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0; the positive solution is x = 4.

The graph of y = −(x − 3)² + 5 has its vertex at which point?

Answer: (3, 5)

In vertex form y = a(x − h)² + k, the vertex is (h, k) = (3, 5).

If x² + bx + 36 = 0 has two equal real roots, which of the following could be a value of b?

Answer: −12

Equal roots: b² − 4(36) = 0 → b² = 144 → b = ±12; b = −12 is listed.

The system y = x + 3 and y = x² − x is solved. What is the sum of the x-values at the intersection points?

Answer: 2

x + 3 = x² − x → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0; x = 3 or x = −1; sum = 2.