SAT Math: Nonlinear Equations (Drill 3)

Question 1. How many real solutions does the system y = x² and y = −4 have?

• A) Zero ✓
• B) One
• C) Two
• D) Infinitely many

Explanation: x² = −4 has no real solutions since x² is always ≥ 0. The parabola y = x² never reaches y = −4.

Question 2. What is the solution to √2x + 3 = x?

• A) x = 3 only ✓
• B) x = −1 only
• C) x = 3 and x = −1
• D) No solution

Explanation: Square both sides: 2x + 3 = x². Rearrange: x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0. x = 3 or x = −1. Check: √9 = 3 ✓ but √1 ≠ −1. Only x = 3 is valid.

Question 3. If 3/(x − 1) = 6, what is the value of x?

• A) 1/2
• B) 3/2 ✓
• C) 2
• D) −1

Explanation: Multiply both sides by (x − 1): 3 = 6(x − 1) = 6x − 6. So 9 = 6x, x = 3/2.

Question 4. The equation x⁴ − 5x² + 4 = 0 can be factored as (x² − 1)(x² − 4) = 0. How many real solutions does it have?

• A) 2
• B) 3
• C) 4 ✓
• D) 1

Explanation: x² − 1 = 0 gives x = ±1. x² − 4 = 0 gives x = ±2. Total: 4 real solutions.

Question 5. For what value of k does the equation x² + 6x + k = 0 have exactly one real solution?

• A) 6
• B) 9 ✓
• C) 12
• D) 3

Explanation: One real solution when discriminant = 0: b² − 4ac = 36 − 4k = 0. k = 9.