Nonlinear Equations and Systems Flashcards

Which of the following is equivalent to x² − 16?

Answer: (x − 4)(x + 4)

x² − 16 is a difference of squares: (x−4)(x+4).

What is the positive solution of 3x² − 27 = 0?

Answer: x = 3

3x² = 27 → x² = 9 → x = ±3; the positive solution is x = 3.

If (x − 3)² = 25, what are the possible values of x?

Answer: x = 8 or x = −2

x − 3 = ±5 → x = 8 or x = −2.

Which quadratic has roots at x = −1 and x = 5?

Answer: x² − 4x − 5

Using factor form: (x+1)(x−5) = x² − 4x − 5.

How many real solutions does x² + 4 = 0 have?

Answer: None

x² = −4 has no real solutions since a square cannot be negative.

The product of two consecutive even integers is 48. Which equation models this situation?

Answer: n² + 2n − 48 = 0

Let the smaller even integer be n; n(n+2) = 48 → n² + 2n − 48 = 0.

The graph of y = x² − 2x − 8 crosses the x-axis at two points. What is the distance between these x-intercepts?

Answer: 6

Factor: (x−4)(x+2) = 0 → x = 4 or x = −2; distance = |4 − (−2)| = 6.