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Nonlinear equations and systems

Solve quadratic, radical, and other nonlinear equations, and count how many real solutions they have.

Advanced MathMath
Practice nonlinear equations and systems free

What the SAT actually asks

You solve equations that are not linear, most often quadratics, and answer questions about their solutions. Tasks include factoring, completing the square, using the quadratic formula, solving radical and rational equations, and solving a system made of one line and one curve. A large group of questions never asks for the solutions at all. Instead it asks how many distinct real solutions exist, or asks for the value of an unknown constant that forces exactly one solution, which points straight at the discriminant.

Advanced Math is about 35 percent of the 44 math questions, and nonlinear equations are usually around 3 to 5 of them.

Where students lose points

How to attack these questions

  1. 1Get the equation to equal zero first, then look for a factorable form before reaching for the quadratic formula.
  2. 2If the question mentions the number of real solutions, compute the discriminant instead of solving.
  3. 3Substitute any solution back into the original equation whenever you squared both sides or cleared a denominator.

Example questions

Written for this skill, with the answer explained. 245 more are in the app.

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Nonlinear equations and systems
Real question

Solve x^2 - 9 = 0.

Easier

The graph of y = f(x) is shown in the xy-plane. For which values of x is f(x) = 2?

  • A) x = 3 only
  • B) x = 1 and x = 5 ✓
  • C) x = -1 and x = 7
  • D) x = 2 only

Why: Draw the horizontal line y = 2 and see where it meets the graph. It meets the graph at the grid points (1, 2) and (5, 2), so f(x) = 2 when x = 1 and x = 5. Answering x = 3 only gives the x-coordinate of the vertex, where f(x) = 6, not 2. Answering x = -1 and x = 7 comes from treating the sides as falling 1 unit per step instead of 2, which gives |x - 3| = 4; the graph drops 2 units for each step of 1, so |x - 3| = 2. Answering x = 2 only misreads the graph, since f(2) = 4.

Medium

If x^2 + 10x + c = (x + 5)^2 - 9 for all values of x, what is the value of c?

  • A) -9
  • B) 16 ✓
  • C) 25
  • D) 34

Why: Expand the right side: (x + 5)^2 - 9 = x^2 + 10x + 25 - 9. Combine the constants: 25 - 9 = 16, so the right side is x^2 + 10x + 16 and c = 16. Answering 25 stops at the square's constant and never subtracts the 9. Answering 34 adds the 9 instead of subtracting it. Answering -9 copies the constant outside the square straight into c.

Harder

For what value of x is (x^2 - 9x + 20)/(x - 4) equal to 0?

  • A) 4
  • B) 5 ✓
  • C) 9
  • D) 20

Why: A fraction is 0 when its numerator is 0 and its denominator is not. The numerator factors as (x - 4)(x - 5), so it is 0 at x = 4 and x = 5. At x = 4 the denominator is also 0, so the expression is undefined there. Only x = 5 makes the expression equal to 0. Answering 9 means you gave the sum of the numerator’s two zeros. Answering 20 means you gave the product of the numerator’s two zeros.

Questions students ask

How do I find how many real solutions a quadratic has?

Compute the discriminant, which is b squared minus 4ac, after writing the equation in the form ax squared plus bx plus c equals zero. A positive discriminant means two distinct real solutions, zero means exactly one, and a negative value means no real solutions.

What is an extraneous solution and when do I get one?

It is a value that appears when you solve but does not satisfy the original equation. They show up after you square both sides of a radical equation or multiply out a denominator. Always substitute your answers back into the original before you choose.

How do I solve a system with one linear and one quadratic equation?

Solve the linear equation for one variable and substitute that expression into the quadratic. You get a single quadratic in one variable, which you solve as usual, then plug each result back to find the matching partner value.

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