Nonlinear Equations and Systems Flashcards

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

Answer: x = ±2

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

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

Answer: y = −(x − 1)² + 4

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

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

Answer: x = −3

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

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

Answer: 6 and 7

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.

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

Answer: 16

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

If f(x) = x² + 2x − 8 and g(x) = x + 4, at what x-values does f(x) = g(x)?

Answer: x = −4 and x = 3

x² + 2x − 8 = x + 4 → x² + x − 12 = 0 → (x + 4)(x − 3) = 0; x = −4 or x = 3.

A ball is thrown upward; its height h(t) = −16t² + 48t + 10 feet. What is the maximum height reached?

Answer: 46 feet

Vertex at t = −48/(2×−16) = 1.5 s; h(1.5) = −16(2.25) + 48(1.5) + 10 = −36 + 72 + 10 = 46 feet.