Nonlinear Equations and Systems Flashcards

If 3x² − 75 = 0, what are the solutions for x?

Answer: x = ±5

3x² = 75 → x² = 25 → x = ±5.

The system y = x² and y = 3x − 2 intersects at two points. What is the larger x-value?

Answer: x = 2

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

For what positive value of x does x² − 5x + 4 = 0?

Answer: x = 1 and x = 4 (both positive)

(x − 1)(x − 4) = 0; both x = 1 and x = 4 are positive solutions.

A parabola opens downward and has x-intercepts at x = −1 and x = 5. Which equation could model this parabola?

Answer: y = −(x + 1)(x − 5)

x-intercepts at −1 and 5 give factors (x + 1)(x − 5); downward opening requires a negative leading coefficient: y = −(x + 1)(x − 5).

The roots of x² − 8x + 15 = 0 are p and q. What is the value of p² + q²?

Answer: 34

p + q = 8; pq = 15; p² + q² = (p + q)² − 2pq = 64 − 30 = 34.

What value of m makes mx² − 6x + 1 = 0 have exactly one real solution?

Answer: m = 9

One real solution when discriminant = 0: (−6)² − 4m(1) = 0 → 36 = 4m → m = 9.

If x² + 5x = −6, what are the solutions?

Answer: x = −2 and x = −3

x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0; x = −2 or x = −3.