Nonlinear Equations and Systems Flashcards

Which of the following is a factor of x² + 7x + 12?

Answer: (x + 3)

x² + 7x + 12 = (x+3)(x+4), so (x+3) is a factor.

The line y = −3x + 5 and the parabola y = x² − 4 intersect at two points. What is the product of the x-coordinates of these two points?

Answer: −9

x² − 4 = −3x + 5 → x² + 3x − 9 = 0; by Vieta's, product of roots = −9/1 = −9.

What is the maximum value of the function f(x) = −x² + 4x + 1?

Answer: 5

Vertex x = −4/(2·(−1)) = 2; f(2) = −4 + 8 + 1 = 5.

The equation kx² − 6x + 1 = 0 has two distinct real solutions. Which of the following must be true?

Answer: k < 9

Discriminant > 0: 36 − 4k > 0 → k < 9.

The roots of ax² + bx + c = 0 are both negative. If a > 0, which of the following must be true?

Answer: b > 0 and c > 0

Sum of roots = −b/a 0; product of roots = c/a > 0 → c > 0.

What is the y-intercept of the parabola y = 2x² − 5x + 3?

Answer: y = 3

Set x = 0: y = 2(0)² − 5(0) + 3 = 3.

Solve the system y = x + 2 and y = x² by substitution. What are the x-values of the solutions?

Answer: x = 2 and x = −1

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