SAT Advanced Math: quadratics, exponentials and nonlinear equations

What each form of a function tells you

Many Advanced Math questions ask which form of an expression shows a particular feature. Learn this table.

Domain share and topic list: College Board Math overview and Assessment Framework for the Digital SAT Suite, checked 6 October 2026.

The key ideas

Quadratics are the core of the domain. You should be able to move between the three forms in the table: factorise to find zeros, complete the square to find the vertex, and expand to compare with a given standard form. The discriminant b^2 - 4ac tells you the number of real solutions: positive means two, zero means one, negative means none. The sum of the solutions of ax^2 + bx + c = 0 is -b/a and their product is c/a, which answers some questions without solving.

Exponential functions have the form a(b)^t. The value a is the starting amount and b is the growth factor per time period. A factor of 1.05 means 5% growth; 0.9 means 10% decay. Linear functions add the same amount each period, exponential functions multiply by the same factor, and SAT questions often ask you to tell them apart from a table.

Radical and rational equations can produce extraneous solutions. When you square both sides of an equation, or multiply by an expression containing x, always substitute your answers back into the original equation and reject any that fail or make a denominator zero.

Equivalent expression questions test structure. Two expressions are equivalent if they are equal for every allowed value of x. When unknown constants are involved, expand and match coefficients: if (ax + 3)(x - 2) = 4x^2 + bx - 6 for all x, then a = 4 and b = 3 - 8 = -5.

Worked example 1: completing the square

Question: The function f(x) = x^2 - 6x + 5 is rewritten in the form f(x) = (x - h)^2 + k. What is the value of k, and what does it show?

1. Halve the x coefficient: -6 / 2 = -3, so the square is (x - 3)^2 = x^2 - 6x + 9.
2. Adjust the constant: x^2 - 6x + 5 = (x - 3)^2 - 9 + 5 = (x - 3)^2 - 4.
3. So h = 3 and k = -4. The vertex is (3, -4), so -4 is the minimum value of f.
4. Check by expanding: (x - 3)^2 - 4 = x^2 - 6x + 9 - 4 = x^2 - 6x + 5. Correct.

Worked example 2: interpreting an exponential model

Question: The number of subscribers to a channel t years after launch is modelled by P(t) = 800(1.05)^t. What does 1.05 mean, and how many subscribers does the model predict after 2 years?

1. 800 is the value at t = 0, the number of subscribers at launch.
2. 1.05 is the growth factor: each year the number is multiplied by 1.05, which is a 5% increase per year.
3. After 2 years: P(2) = 800 × 1.05^2 = 800 × 1.1025 = 882.
4. Answer: 5% annual growth, and 882 subscribers after 2 years. Note it is not 800 + 2 × 40 = 880; exponential growth compounds.

Worked example 3: a radical equation with an extraneous root

Question: What value of x satisfies sqrt(x + 7) = x - 5?

1. Square both sides: x + 7 = (x - 5)^2 = x^2 - 10x + 25.
2. Rearrange: 0 = x^2 - 11x + 18.
3. Factorise: (x - 9)(x - 2) = 0, so x = 9 or x = 2.
4. Check x = 9: sqrt(16) = 4 and 9 - 5 = 4. It works.
5. Check x = 2: sqrt(9) = 3 but 2 - 5 = -3. A square root is never negative, so x = 2 is extraneous.
6. Answer: x = 9 only.

Worked example 4: sum of solutions without solving

Question: What is the sum of the solutions of 2x^2 - 7x + 3 = 0?

1. For ax^2 + bx + c = 0, the sum of the solutions is -b/a.
2. Here a = 2 and b = -7, so the sum is 7/2.
3. Check by solving: (2x - 1)(x - 3) = 0 gives x = 1/2 and x = 3, and 1/2 + 3 = 7/2.
4. On a typed-answer question, enter 7/2 or 3.5.

Common mistakes that cost marks

• Giving the x-coordinate of the vertex when the question asks for the minimum or maximum value.
• Reading the growth rate as 1.05% instead of 5%, or reading 0.9 as 90% decay instead of 10%.
• Forgetting to check for extraneous roots after squaring.
• Sign errors in vertex form: (x + 3)^2 has its vertex at x = -3, not 3.
• Treating exponential growth as linear when estimating values over several periods.
• Matching only one coefficient in equivalent-expression questions instead of all of them.

Exam technique and how a tutor helps

Advanced Math questions are where Desmos pays off most: graphing a quadratic shows its zeros and vertex, and graphing both sides of a radical equation shows immediately that only one solution exists. Use it to check structure-based answers too. But questions with unknown constants, such as 'for which value of c does the equation have no real solutions', still need the discriminant or coefficient matching.

Questions in each module run from easier to harder, so the toughest Advanced Math items tend to sit near the end of a module. If your goal is a Math score in the high 600s or above, this is the domain to master, because the hardest questions in it reward knowing the forms and rules rather than long calculation.

In one-to-one lessons a tutor diagnoses which of the four question families (forms of quadratics, exponential models, extraneous roots, equivalent expressions) costs the student marks, teaches the rule behind it on the shared whiteboard, and then mixes it into timed sets so the student learns to recognise it under pressure. Students from IGCSE and IB often need extra practice reading exponential models in American context language.

Self-check: can you do these?

• Solve x^2 - 2x - 15 = 0. (Answer: x = 5 or x = -3)
• Write x^2 + 10x + 21 in vertex form. (Answer: (x + 5)^2 - 4)
• A(t) = 500(0.9)^t. Describe the change per year. (Answer: decreases by 10% per year)
• Solve sqrt(2x + 3) = x. (Answer: x = 3; x = -1 is extraneous)
• How many real solutions does 3x^2 + 2x + 5 = 0 have? (Answer: none, the discriminant is -56)