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Solving
Nonlinear Functions

Master the techniques for finding solutions to quadratic, exponential, and radical equations: the key nonlinear function types on the SAT.

SAT Math

What Is a Nonlinear Function?

A nonlinear function is any function whose graph is not a straight line. The variable appears with an exponent other than 1, inside a radical, or as the base/exponent of an exponential expression.

Quadratic
\( f(x) = ax^2 + bx + c \)
Exponential
\( f(x) = a \cdot b^x \)
Radical
\( f(x) = \sqrt{ax + b} \)

Visualizing Nonlinear Graphs

x y Quadratic Exponential Radical

Why This Matters on the SAT

Nonlinear equations appear in roughly 30% of SAT Math questions. You will be asked to find zeros, intersections, and solution sets. Speed and accuracy depend on recognizing which technique to use.

1
Quadratics are the most common: factoring, completing the square, and the quadratic formula.
2
Exponentials appear in growth/decay word problems and system-of-equations contexts.
3
Radicals and absolute values require careful isolation and checking for extraneous solutions.

Core Rule: Setting \( f(x) = 0 \)

To solve a nonlinear function, set it equal to zero and find the \( x \)-values (called roots, zeros, or solutions) that satisfy the equation.

x y root root vertex -3 3 0

The Quadratic Formula

For any quadratic equation \( ax^2 + bx + c = 0 \), the solutions are given by:

\( b^2 - 4ac > 0 \)
Two distinct real roots
\( b^2 - 4ac = 0 \)
One repeated real root
\( b^2 - 4ac < 0 \)
No real roots

Factoring vs. Formula

Factoring Quadratic Formula
Speed Fast when factors are integers Slower but always works
Best for \( x^2 + 5x + 6 = 0 \) \( 3x^2 - 7x + 1 = 0 \)
Requires Recognizing integer factor pairs Memorizing the formula
SAT tip Try factoring first (30 seconds) Fall back to this if factoring fails

Beyond Quadratics: Exponential and Radical

Exponential Equations
Isolate the exponential term, then use logarithms or match bases.
Example: \( 2^x = 16 \Rightarrow x = 4 \)
Radical Equations
Isolate the radical, square both sides, then check for extraneous solutions.
Example: \( \sqrt{x+3} = 5 \Rightarrow x = 22 \)

How to Solve Step by Step

1
Identify the type: Is it quadratic, exponential, radical, or a system?
2
Rearrange to standard form: Move all terms to one side so you have \( f(x) = 0 \) (or isolate the nonlinear term).
3
Choose your method: Factoring, quadratic formula, matching bases, squaring both sides, or substitution.
4
Solve for \( x \): Execute the technique and simplify.
5
Check your answer: Substitute back into the original equation, especially for radical and rational equations.

Example 1: Factoring

Solve: \( x^2 - 5x + 6 = 0 \)

1
Identify: This is a quadratic in standard form \( ax^2 + bx + c = 0 \).
2
Find factors: We need two numbers that multiply to 6 and add to \(-5\): those are \(-2\) and \(-3\).
3
Factor: \( (x - 2)(x - 3) = 0 \)
4
Solve: \( x - 2 = 0 \Rightarrow x = 2 \) or \( x - 3 = 0 \Rightarrow x = 3 \)

Example 2: Quadratic Formula

Solve: \( 2x^2 + 3x - 5 = 0 \)

1
Identify coefficients: \( a = 2 \), \( b = 3 \), \( c = -5 \).
2
Discriminant: \( b^2 - 4ac = 9 - 4(2)(-5) = 9 + 40 = \) \(49\)
3
Apply formula: \( x = \dfrac{-3 \pm \sqrt{49}}{2(2)} = \dfrac{-3 \pm 7}{4} \)
4
Two solutions: \( x = \dfrac{-3 + 7}{4} = 1 \) or \( x = \dfrac{-3 - 7}{4} = -\dfrac{5}{2} \)

Example 3: Radical Equation

Solve: \( \sqrt{2x + 3} = x \)

1
Square both sides: \( 2x + 3 = x^2 \)
2
Rearrange: \( x^2 - 2x - 3 = 0 \)
3
Factor: \( (x - 3)(x + 1) = 0 \), so \( x = 3 \) or \( x = -1 \).
4
Check: \( x = 3 \): \( \sqrt{9} = 3 \) ✓. \( x = -1 \): \( \sqrt{1} = 1 \neq -1 \) ✕ extraneous.

Systems: Linear Meets Nonlinear

The SAT often asks: at what point(s) does a line intersect a parabola? Substitute the linear expression into the quadratic and solve.

x y (0, 0) (2, 4)
\( y = x^2 \)
\( y = 2x \)

Completing the Square

This method rewrites a quadratic into vertex form \( a(x - h)^2 + k = 0 \), making it easy to solve and also revealing the vertex.

1
Start: \( x^2 + 6x + 2 = 0 \). Move the constant: \( x^2 + 6x = -2 \).
2
Half of \( b \), squared: \( \left(\frac{6}{2}\right)^2 = 9 \). Add to both sides: \( x^2 + 6x + 9 = 7 \).
3
Factor: \( (x + 3)^2 = 7 \).
4
Solve: \( x + 3 = \pm\sqrt{7} \), so \( x = -3 \pm \sqrt{7} \).

Common Mistakes

✕ Sign Errors in the Formula
Forgetting the negative sign on \( b \) or mishandling \( \pm \). Write out each step carefully.
✕ Forgetting Extraneous Solutions
Squaring both sides of a radical equation introduces false roots. Always substitute back.
✕ Not Setting Equal to Zero
Trying to factor \( x^2 + 3x = 10 \) as-is. You must first write \( x^2 + 3x - 10 = 0 \).

Summary of Methods

Equation Type Primary Method Backup Method Key Check
Quadratic Factoring Quadratic Formula Discriminant sign
Radical Isolate, square Graphing Extraneous roots
Exponential Match bases Logarithms Domain restrictions
System (linear + quad) Substitution Graphing Number of intersections

Question 1 of 5

What are the solutions to \( x^2 - 7x + 12 = 0 \)?

Question 2 of 5

What is the discriminant of \( 3x^2 - 4x + 5 = 0 \), and how many real solutions does it have?

Question 3 of 5

Solve: \( \sqrt{3x + 1} = x - 1 \). Which value(s) of \( x \) are valid?

Question 4 of 5

At what \( x \)-values does the line \( y = x + 2 \) intersect the parabola \( y = x^2 \)?

Question 5 of 5

If \( 2^{x+1} = 32 \), what is the value of \( x \)?