Advanced Math · Math
Nonlinear equations and systems
Nonlinear equations and systems: quadratics, radicals, rational equations, and counting the intersections of a line and a parabola.
51 questions in the bank · 4 easier · 25 medium · 22 harder · this domain is ≈35% of the section
What to watch for
Squaring both sides can invent solutions, so check candidates in the original equation. For intersections, the discriminant of the combined equation tells you how many there are.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- If x² = 36 and x > 0, what is the value of x?
- If x² = 49 and x > 0, what is the value of x?
- If x² = 81 and x > 0, what is the value of x?
- If x² = 144 and x > 0, what is the value of x?
Worked examples
easyIf x² = 36 and x > 0, what is the value of x?
If x² = 36 and x > 0, what is the value of x?
- A. 6
- B. 36
- C. −6
- D. 18
Answer: A. 6
Explanation
Take the square root of both sides: x = ±√36 = ±6. The condition x > 0 selects the positive root, so x = 6. Without that condition, a quadratic like this has two solutions — the SAT often uses the constraint to pick one.
mediumIf the solutions to the equation x² − 7x + 12 = 0 are r and s, what is the val…
If the solutions to the equation x² − 7x + 12 = 0 are r and s, what is the value of r + s?
- A. −7
- B. 7
- C. 8
- D. 12
Answer: B. 7
Explanation
Factor: x² − 7x + 12 = (x − 3)(x − 4), so the solutions are 3 and 4, and r + s = 7. Shortcut worth knowing: for x² + bx + c = 0, the roots sum to −b and multiply to c.
hardWhat is the solution to the equation √(x + 16) = x − 4? (If there is more than…
What is the solution to the equation √(x + 16) = x − 4? (If there is more than one value, enter the one that satisfies the original equation.)
Answer: 9
Explanation
Square both sides: x + 16 = (x − 4)², i.e. x² − 8x + 16 on the right. Rearranged, the quadratic's candidates must be checked in the ORIGINAL equation, because squaring can invent solutions. Checking shows x = 9 works (√25 = 5 = 9 − 4).