Nonlinear Equations and Systems Flashcards

The function g(x) = x² − 4x + 3 is reflected over the x-axis to form h(x). What is h(5)?

Answer: −8

h(x) = −g(x) = −(x² − 4x + 3); h(5) = −(25 − 20 + 3) = −8.

The parabola y = x² + 2x − 15 is tangent to the horizontal line y = k. What is the value of k?

Answer: −16

Vertex y-value = minimum = c − b²/(4a) = −15 − 4/4 = −15 − 1 = −16.

Solve for x: (x − 4)² − 9 = 0

Answer: x = 1 and x = 7

(x − 4)² = 9 → x − 4 = ±3 → x = 7 or x = 1.

For the system y = x² + 1 and y = 5, how many solutions are there?

Answer: 2

x² + 1 = 5 → x² = 4 → x = ±2; two solutions.

The equation 4x² − 4x + 1 = 0 has how many distinct real solutions?

Answer: 1

Discriminant = (−4)² − 4(4)(1) = 16 − 16 = 0; exactly one real solution (a repeated root).

Which equation has roots at x = −2 and x = 5?

Answer: x² − 3x − 10 = 0

(x + 2)(x − 5) = x² − 3x − 10 = 0.

Which of the following is the vertex form of y = 2x² − 12x + 10?

Answer: y = 2(x − 3)² − 8

Complete the square: 2(x² − 6x) + 10 = 2(x² − 6x + 9 − 9) + 10 = 2(x − 3)² − 18 + 10 = 2(x − 3)² − 8.