Nonlinear Equations and Systems Flashcards

The product of two numbers is 40 and their sum is 13. Which quadratic equation has these two numbers as roots?

Answer: x² − 13x + 40 = 0

By Vieta's formulas: sum = 13 and product = 40 → x² − 13x + 40 = 0.

What is the x-coordinate of the axis of symmetry of y = −2x² + 8x − 5?

Answer: x = 2

Axis of symmetry: x = −b/(2a) = −8 / (2·(−2)) = −8/(−4) = 2.

The system y = 2x + k and y = x² has exactly one solution. What is k?

Answer: −1

x² = 2x + k → x² − 2x − k = 0; one solution when discriminant = 4 + 4k = 0 → k = −1.

If 5x² = 45, what is the value of x²?

Answer: 9

x² = 45 ÷ 5 = 9.

Which of the following best describes the solutions of x² + 2x + 5 = 0?

Answer: No real solutions

Discriminant = 4 − 20 = −16 < 0, so there are no real solutions.

The parabola y = x² − 6x + 5 is shifted 3 units to the right. What is the equation of the new parabola?

Answer: y = x² − 12x + 32

Replace x with (x−3): (x−3)² − 6(x−3) + 5 = x² − 6x + 9 − 6x + 18 + 5 = x² − 12x + 32.

For the equation 2x² + 5x − 3 = 0, what is the product of the roots?

Answer: −3/2

By Vieta's formulas, product of roots = c/a = −3/2.