Nonlinear Equations and Systems Flashcards

For the quadratic f(x) = 2x² − 8x + 6, what are the x-intercepts?

Answer: x = 1 and x = 3

2x² − 8x + 6 = 2(x² − 4x + 3) = 2(x − 1)(x − 3); x-intercepts at x = 1 and x = 3.

Solve: x² − 81 = 0

Answer: x = ±9

x² = 81; x = ±9.

The system y = 2x − 1 and y = x² − 4x + 3 is solved. What is the product of the x-coordinates of the solutions?

Answer: 4

2x − 1 = x² − 4x + 3 → x² − 6x + 4 = 0; by Vieta's, product of roots = c/a = 4.

A quadratic has a positive leading coefficient and vertex at (2, −3). How many x-intercepts does it have?

Answer: 2

The parabola opens upward (positive leading coefficient) and its minimum is −3 < 0; it must cross the x-axis twice.

If x² + 12x + c = (x + 6)², what is the value of c?

Answer: 36

(x + 6)² = x² + 12x + 36, so c = 36.

The equation 5x² = 3x has a non-zero solution. What is it?

Answer: x = 3/5

5x² − 3x = 0 → x(5x − 3) = 0; solutions are x = 0 and x = 3/5; the non-zero solution is x = 3/5.

The parabola y = x² − 4x + 3 is shifted 2 units to the right. What is the new equation?

Answer: y = x² − 8x + 15

Shifting right by 2 replaces x with (x − 2): y = (x−2)² − 4(x−2) + 3 = x² − 4x + 4 − 4x + 8 + 3 = x² − 8x + 15.