Nonlinear Equations and Systems Flashcards

What are the solutions to x² − 5x + 6 = 0?

Answer: x = 2 and x = 3

Factor as (x−2)(x−3) = 0, giving x = 2 and x = 3.

The quadratic equation x² − 4x + k = 0 has exactly one real solution. What is the value of k?

Answer: 4

For exactly one solution the discriminant = 0: (−4)² − 4(1)(k) = 0 → 16 = 4k → k = 4.

Which of the following are both solutions to 2x² + 3x − 2 = 0?

Answer: x = 1/2 and x = −2

Factor: (2x−1)(x+2) = 0 → x = 1/2 or x = −2.

How many real solutions does the equation x² + 6x + 10 = 0 have?

Answer: None

Discriminant = 6² − 4(1)(10) = 36 − 40 = −4 < 0, so no real solutions exist.

What is the vertex of the parabola y = x² − 8x + 15?

Answer: (4, −1)

x-vertex = −(−8)/(2·1) = 4; y = 16 − 32 + 15 = −1; vertex = (4, −1).

Solve for x: x² = 49

Answer: x = 7 or x = −7

Taking the square root of both sides gives x = ±7.

The parabola y = x² + bx + c passes through (0, 5) and (5, 0). What is the value of b?

Answer: −6

From (0,5): c = 5. From (5,0): 25 + 5b + 5 = 0 → 5b = −30 → b = −6.

Which values of x satisfy x² − x − 12 = 0?

Answer: x = 4 and x = −3

Factor: (x−4)(x+3) = 0 → x = 4 or x = −3.

For the equation 3x² − 12x + 12 = 0, what is the value of the discriminant?

Answer: 0

Discriminant = (−12)² − 4(3)(12) = 144 − 144 = 0.

Using the quadratic formula, what are the solutions of x² − 6x + 7 = 0?

Answer: x = 3 ± √2

x = [6 ± √(36−28)] / 2 = [6 ± √8] / 2 = [6 ± 2√2] / 2 = 3 ± √2.

The system y = x² and y = x + 6 is solved by substitution. What are the x-values of the intersection points?

Answer: x = 3 and x = −2

Set x² = x + 6 → x² − x − 6 = 0 → (x−3)(x+2) = 0 → x = 3 or x = −2.

How many points of intersection does the system y = x² − 3 and y = 2x have?

Answer: 2

x² − 3 = 2x → x² − 2x − 3 = 0; discriminant = 4 + 12 = 16 > 0, so 2 intersections.

The function f(x) = x² − 4x + 3. At which x-value does f(x) reach its minimum?

Answer: x = 2

The vertex x-coordinate = 4 / (2·1) = 2; this is the minimum since a > 0.

The system y = x + 2 and y = x² − 4 has two solutions. What is the sum of their x-coordinates?

Answer: 1

x² − 4 = x + 2 → x² − x − 6 = 0; by Vieta's, sum of roots = −(−1)/1 = 1.

Using the quadratic formula, what are the solutions to 2x² − 4x − 6 = 0?

Answer: x = 3 and x = −1

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

The parabola y = (x − 3)² − 4 intersects the x-axis at which two values of x?

Answer: x = 1 and x = 5

Set y = 0: (x−3)² = 4 → x−3 = ±2 → x = 5 or x = 1.

How many solutions does the system y = x² + 3x + 2 and y = x + 2 have?

Answer: 2

x² + 3x + 2 = x + 2 → x² + 2x = 0 → x(x+2) = 0 → x = 0 or x = −2; two solutions.

The equation x² + px + 16 = 0 has two equal real roots. What is the positive value of p?

Answer: 8

Equal roots require discriminant = 0: p² − 64 = 0 → p = ±8; the positive value is 8.

A parabola y = x² − 4x + c does not intersect the x-axis. Which value of c satisfies this condition?

Answer: c = 5

Discriminant 4; only c = 5 satisfies this.

If x² − 7x + 10 = 0, what is the value of x² − 7x?

Answer: −10

From the equation: x² − 7x + 10 = 0 → x² − 7x = −10.