Nonlinear Equations and Systems Flashcards

A quadratic equation has roots x = 2 + √3 and x = 2 − √3. What is the quadratic equation?

Answer: x² − 4x + 1 = 0

Sum = 4, product = (2)² − (√3)² = 4 − 3 = 1; equation: x² − 4x + 1 = 0.

The height of a projectile is modeled by h = −5t² + 20t + 60. At what positive time t does the projectile hit the ground (h = 0)?

Answer: t = 6 only

−5t² + 20t + 60 = 0 → t² − 4t − 12 = 0 → (t−6)(t+2) = 0 → t = 6 (positive) or t = −2 (rejected).

If 2x² − 8 = 0, what are the solutions for x?

Answer: x = ±2

2x² = 8 → x² = 4 → x = ±2.

The system y = x² − x − 6 and y = 2 is solved. What is the sum of the x-coordinates of the intersection points?

Answer: 1

x² − x − 6 = 2 → x² − x − 8 = 0; by Vieta's formulas, sum of roots = −(−1)/1 = 1.

What value of c makes x² + 10x + c a perfect square trinomial?

Answer: 25

For a perfect square: c = (b/2)² = (10/2)² = 25.

For the system y = x² − 1 and y = 3x − 3, at which x-values do the graphs intersect?

Answer: x = 1 and x = 2

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

The quadratic y = −x² + 6x − 5 has a maximum value. What is that maximum value?

Answer: 4

Vertex x = 6/(2·1) = 3; y = −9 + 18 − 5 = 4.