Drill 4 · Math · Nonlinear Equations
SAT Math: Nonlinear Equations (Drill 4) is a Math practice drill covering Nonlinear Equations. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.
SAT nonlinear equation questions involve solving quadratics by factoring or the quadratic formula, radical equations requiring extraneous solution checks, rational equations, substitution in higher-degree equations, and using the discriminant to find special solution conditions.
Question 1. If x² − 9 = 0, what is the sum of all possible values of x?
Explanation: x² − 9 = 0 factors as (x − 3)(x + 3) = 0, so x = 3 or x = −3. The sum is 3 + (−3) = 0.
Question 2. What are all the solutions to √3x + 7 = x + 1?
Explanation: Square both sides: 3x + 7 = (x + 1)² = x² + 2x + 1. Rearrange: x² − x − 6 = 0 → (x − 3)(x + 2) = 0. So x = 3 or x = −2. Check: √16 = 4 and 3 + 1 = 4 ✓. But √1 = 1 and −2 + 1 = −1 ✗. Only x = 3 is valid.
Question 3. If 5/(x + 2) + 1 = 3, what is the value of x?
Explanation: Subtract 1 from both sides: 5/(x + 2) = 2. Multiply both sides by (x + 2): 5 = 2(x + 2) = 2x + 4. So 1 = 2x, and x = 1/2.
Question 4. The equation x⁴ − 13x² + 36 = 0 can be solved by substitution. How many distinct real solutions does it have?
Explanation: Let u = x². Then u² − 13u + 36 = 0 → (u − 4)(u − 9) = 0. So u = 4 or u = 9. Then x² = 4 gives x = ±2, and x² = 9 gives x = ±3. That is 4 distinct real solutions.
Question 5. For what value of c does the equation x² − 10x + c = 0 have exactly one real solution?
Explanation: Exactly one real solution occurs when the discriminant equals zero: b² − 4ac = (−10)² − 4(1)(c) = 100 − 4c = 0. Solving gives c = 25.