Nonlinear Equations and Systems Flashcards

The sum of a number and its square is 56. Which equation models this situation?

Answer: x² + x − 56 = 0

Let the number be x; x + x² = 56 rearranges to x² + x − 56 = 0.

For the equation 4x² − 20x + 25 = 0, how many real solutions are there?

Answer: 1

Discriminant = (−20)² − 4(4)(25) = 400 − 400 = 0, so exactly one repeated real solution.

What is the solution set of x² − 9 = 0?

Answer: {3, −3}

x² = 9 → x = ±3.

The system y = x² − 2x − 3 and y = 5 is solved. What are the x-values of the intersections?

Answer: x = 4 and x = −2

x² − 2x − 3 = 5 → x² − 2x − 8 = 0 → (x−4)(x+2) = 0 → x = 4 or x = −2.

A ball is thrown upward; its height in feet at time t seconds is h(t) = −16t² + 64t. At what time does the ball return to the ground (h = 0, t > 0)?

Answer: t = 4 s

−16t² + 64t = 0 → −16t(t−4) = 0 → t = 0 or t = 4; the ball returns at t = 4 s.

If the graph of y = x² + bx + 4 touches the x-axis at exactly one point, which of the following could be a value of b?

Answer: b = 4

For one x-intercept, discriminant = 0: b² − 16 = 0 → b = ±4; so b = 4 works.

Solve: (x + 2)(x − 5) = 0

Answer: x = −2 and x = 5

Set each factor to zero: x + 2 = 0 → x = −2; x − 5 = 0 → x = 5.