what is a nonlinear equation?

A nonlinear equation in one variable has powers, roots, or functions of $x$ beyond just $x$ to the first power. Nonlinear equations can curve, bend, or even loop on a graph—and often have more than one solution.

• Quadratic: includes terms like $x^2$ (e.g., $x^2 - 5x + 6 = 0$)
• Cubic: includes $x^3$
• Radical: includes square roots, like $\sqrt{x}$
• Rational: variable in the denominator, like $\frac{1}{x}$
• Absolute value: includes bars, like $|x|$

SAT tip: Nonlinear equations can have 0, 1, 2, or even more real solutions—always check if your answers actually work!

solving nonlinear equations: strategies

Nonlinear equations include quadratics, rationals, radicals, and absolute values. Each one has its own best solving method. Here’s how to tackle each, with an example for every type:

• Quadratics:
  Try factoring first, or use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. You can also complete the square if needed.
  Example: $x^2 - 5x + 6 = 0$
  Factor: $(x-2)(x-3)=0$. Solutions: $x=2$ and $x=3$.
• Rational:
  Multiply both sides by the common denominator to clear fractions. Be careful—always check your answers in the original equation to avoid extraneous solutions.
  Example: $\frac{2}{x} = x - 1$
  Multiply both sides by $x$: $2 = x(x - 1)$
  Rearrange: $x^2 - x - 2 = 0$
  Factor: $(x-2)(x+1)=0$. So $x=2$ or $x=-1$.
  Plug back in: $x=-1$ makes denominator zero (not allowed). Only $x=2$ is valid.
• Radicals:
  Isolate the radical, then square both sides. Always check for “impossible” answers (extraneous roots) at the end.
  Example: $\sqrt{x + 5} = x - 1$
  Square both sides: $x + 5 = (x - 1)^2$
  Expand: $x + 5 = x^2 - 2x + 1$
  Rearrange: $0 = x^2 - 3x - 4$
  Factor: $(x-4)(x+1)=0$
  So $x=4$ or $x=-1$
  Check: $x=-1$ gives $\sqrt{-1 + 5} = -1 - 1$, which is $2 = -2$ (not true). Only $x=4$ works.
• Absolute value:
  Set up two cases: one positive, one negative, then solve both.
  Example: $|2x-5| = 7$
  Two cases: $2x-5=7$ and $2x-5=-7$
  $2x=12$ so $x=6$, and $2x=-2$ so $x=-1$

SAT tip: For quadratics, try factoring before reaching for the formula. For rational and radical equations, always check your solutions in the original equation—some answers might not actually work!

systems: lines and curves

A system of nonlinear equations includes at least one equation that isn’t linear—like a quadratic, cubic, or a radical. On the SAT, the classic case is a line and a parabola (for example, $y = x^2$ and $y = 3x - 2$). Solutions to these systems are simply the points where the two graphs cross—sometimes two, sometimes one, or sometimes none.

• Nonlinear system: At least one equation has $x^2$, $xy$, etc.
• SAT classic: Line and parabola, or two parabolas.
• Solution: Points where the graphs intersect. There can be 0, 1, 2, or even more real solutions.

SAT tip: For systems involving a line and a curve, always set the equations equal and solve the resulting nonlinear equation. You may need to factor, complete the square, or use the quadratic formula.

number of solutions & extraneous roots

Nonlinear equations and systems can have two, three, many, one, or even no real solutions. When solving, especially with radicals or rational expressions, always check for extraneous roots—answers that work in the algebra but not in the original equation!

• Some nonlinear equations or systems have more than two solutions—or sometimes none!
• Always check for extraneous roots when you square both sides or multiply both sides by a variable expression (especially with radicals or rationals).

SAT tip: Every time you square both sides or multiply by a variable, always plug your answers back into the original equation. Don’t get tricked by extraneous roots!

Let's practice with a few College Board questions!