Quadratics, Exponents & Nonlinear: The Curves That Show Up On the SAT

Why "advanced math" sounds scarier than it is

"Advanced Math" on the digital SAT really means: quadratics, exponents, polynomials, exponentials, and the occasional rational expression. None of it is calculus. All of it follows a small handful of patterns that repeat. The trick is recognizing which pattern you're looking at within five seconds.

Quadratics: three forms, three personalities

The SAT picks the form that makes the answer easiest, then asks for the feature stored in a different form. Your skill is converting.

• Standard to factored: factor or use the quadratic formula.
• Standard to vertex: complete the square, or use h = -b/(2a) and plug in for k.
• Factored to standard: just multiply out (FOIL).

The quadratic formula, and when not to use it

x = [-b ± √(b² - 4ac)] / (2a). Yes, memorize it. But on the SAT, you'll often save time by:

1. Factoring when the numbers are friendly. ax² + bx + c with a = 1: find two numbers that multiply to c and add to b.
2. Graphing in Desmos. Type the quadratic, click the x-intercepts, read off the roots. Faster than the formula in many cases.
3. Vieta's shortcut. For ax² + bx + c, sum of roots = -b/a, product of roots = c/a. Sometimes the question only wants the sum or product, no need to find the roots themselves.

The discriminant: how many real solutions?

Inside the quadratic formula sits the discriminant, b² - 4ac. It tells you the number of real solutions:

• Positive: two real solutions.
• Zero: exactly one (a "double root", the parabola just touches the x-axis).
• Negative: no real solutions (the parabola doesn't cross the x-axis).

The SAT loves discriminant questions phrased like: "For what value of k does the equation have exactly one solution?" Set b² - 4ac = 0 and solve for k.

Exponents: seven rules that cover almost everything

1. x^a · x^b = x^a+b
2. x^a / x^b = x^a-b
3. (x^a)^b = x^ab
4. x⁰ = 1 (for any nonzero x)
5. x^-a = 1/x^a
6. x^1/n = ⁿ√x (the nth root)
7. x^a/b = ᵇ√(x^a)

The most common SAT trap: students add exponents when they should multiply, or distribute exponents over a sum. Reminder: (x + y)² is not x² + y². It's x² + 2xy + y².

Exponential functions: the doubling-tripling family

An exponential function looks like f(x) = a · b^x. Here:

• a is the starting value (when x = 0).
• b is the growth or decay factor per unit of x.

If b > 1, growth. If 0 < b < 1, decay. The SAT word problem version: "a population doubles every 7 years, starting at 1,000." That's f(x) = 1,000 · 2^x/7. Notice the x is divided by the doubling period.

Polynomials and end behavior

For higher-degree polynomials, the SAT mostly cares about three things:

• Roots / zeros / x-intercepts. Set y = 0 and factor.
• End behavior. The leading term controls the ends. x⁴ goes up on both sides; -x³ goes up on the left, down on the right.
• Multiplicity. A factor like (x - 2)² means the graph touches the x-axis at x = 2 and bounces, instead of crossing.

Rational expressions: a small but reliable category

For (x² - 9) / (x - 3), factor the top: (x - 3)(x + 3) / (x - 3) = x + 3, with x ≠ 3. Watch the SAT trap of forgetting the excluded value.

A worked example pulling it together

"The function f(x) = 3x² + bx + 12 has exactly one real solution. What is the value of b² ?"

Discriminant condition: b² - 4ac = 0. Here a = 3, c = 12. So b² - 4(3)(12) = 0, which gives b² = 144.

You didn't need to solve for b itself. The question asked for b². Read the question twice. The SAT often hands you a small gift this way.

How to practice nonlinear math

1. For every quadratic question, name the form: standard, factored, or vertex. Then ask: which form would make the question trivial? Convert and answer.
2. Drill exponent rules with mixed problems until they're automatic. About 15 minutes a day for a week is enough.
3. Use Desmos to verify your algebra. If your answer doesn't match the graph, find the error before you move on.