Nonlinear equations and systems

Digital SAT · Math · Advanced Math. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A quadratic equation can be written ax² + bx + c = 0. Move everything to one side first. Then factor and set each factor equal to 0, or use the quadratic formula. The discriminant, b² − 4ac, tells you how many real solutions there are without solving.

For an equation with a square root, isolate the root and square both sides. For an equation with fractions, multiply through by the common denominator. Both methods can create extraneous solutions, so test every answer in the original equation.

For an absolute value equation, split it into two cases. For a line and a curve together, substitute the linear equation into the other one.

Rules to remember

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a)
b² − 4ac > 0: two real solutions; = 0: exactly one; < 0: no real solution.
For ax² + bx + c = 0: sum of solutions = −b/a, product of solutions = c/a.
If (x − p)(x − q) = 0, then x = p or x = q.
|A| = k with k ≥ 0 means A = k or A = −k; if k < 0 there is no solution.
A value that makes a denominator 0 can never be a solution.
A square root sign means the non-negative root, so √(…) cannot equal a negative number.

Common mistake

Keeping every answer after squaring both sides. Squaring can add a false solution, so substitute each value back into the original equation and reject any that fails.

Worked example

√(2x + 3) = x. What is the solution to the equation?

Square both sides: 2x + 3 = x², so x² − 2x − 3 = 0.
Factor: (x − 3)(x + 1) = 0, so x = 3 or x = −1.
Check x = 3: √(6 + 3) = √9 = 3. True.
Check x = −1: √(−2 + 3) = √1 = 1, which is not −1. Reject.

Answer: x = 3

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