Nonlinear Equations and Systems Flashcards

If f(x) = x² − 4 and g(x) = x + 2, for what values of x does f(x) = g(x)?

Answer: x = −2 and x = 3

x² − 4 = x + 2 → x² − x − 6 = 0 → (x − 3)(x + 2) = 0; x = 3 or x = −2.

The solutions of x² + px + q = 0 are x = 4 and x = −7. What is the value of p?

Answer: p = 3

Sum of roots = 4 + (−7) = −3 = −p, so p = 3.

The parabola y = x² − 2x − 3 is shifted 4 units upward. How many x-intercepts does the new parabola have?

Answer: 1

New equation: y = x² − 2x + 1 = (x − 1)²; this touches the x-axis at exactly one point.

What is the product of all real solutions to x³ − 4x = 0?

Answer: 0

x(x² − 4) = x(x − 2)(x + 2) = 0; solutions are x = 0, 2, −2; product = 0 × 2 × (−2) = 0.

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

Answer: −1

Set x² = 2x + k → x² − 2x − k = 0; tangency requires discriminant = 0: 4 + 4k = 0 → k = −1.

The parabola y = x² − 6x + 9 is the graph of which factored form?

Answer: y = (x − 3)²

x² − 6x + 9 = (x − 3)².

The quadratic y = ax² + bx + c has a y-intercept of 6 and passes through (1, 9) and (−1, 7). What is the value of a?

Answer: 2

c = 6; (1,9): a + b + 6 = 9 → a + b = 3; (−1,7): a − b + 6 = 7 → a − b = 1; adding: 2a = 4 → a = 2.