Nonlinear Equations and Systems Flashcards

What is the sum of the roots of the equation 3x² − 12x + 7 = 0?

Answer: 4

By Vieta's formulas, sum of roots = −b/a = −(−12)/3 = 4.

Which of the following equations has no real solutions?

Answer: x² + 3x + 10 = 0

For x² + 3x + 10 = 0, discriminant = 9 − 40 = −31 < 0, so no real solutions.

If f(x) = x² − 6x + 8, at which x-value does the function reach its minimum?

Answer: x = 3

The vertex x-coordinate = −b/(2a) = 6/2 = 3.

The parabola y = x² + 4x − 12 crosses the x-axis at two points. What is the product of those x-intercepts?

Answer: −12

By Vieta's formulas, product of roots = c/a = −12/1 = −12.

Which of the following is a correct factored form of 6x² + 11x − 10?

Answer: (2x + 5)(3x − 2)

(2x + 5)(3x − 2) = 6x² − 4x + 15x − 10 = 6x² + 11x − 10. ✓

The system y = x² − 5x + 6 and y = 0 is solved. What is the positive difference between the two solutions?

Answer: 1

x² − 5x + 6 = (x − 2)(x − 3) = 0; roots are x = 2 and x = 3; difference = 1.

If (2x − 1)² = 49, what are the two values of x?

Answer: x = 4 and x = −3

2x − 1 = ±7; case 1: 2x = 8, x = 4; case 2: 2x = −6, x = −3.