SAT Advanced Math: method and traps

Advanced Math is where the SAT stops asking you to solve and starts asking you to recognise. The arithmetic is rarely harder than in Algebra; what changes is that the fastest route depends on spotting the shape of the expression before you touch it.

Practise the 112 questions Play a timed round

This area at a glance

6 worked examples · 6 traps

112

questions

  • Foundation 14
  • Standard 71
  • Challenge 27

What this area asks for

  • Quadratic equations and their solutions
  • Vertex form and the shape of a parabola
  • Factoring, including differences of squares
  • Polynomial roots and factors
  • Exponent rules and exponential equations
  • Rational expressions

The method

One routine that applies across the whole area, rather than a rule for each question type.

  1. 1

    Try factoring before the quadratic formula

    Most quadratics on this test are built to factor. Look for two numbers that multiply to the constant term and add to the middle coefficient. If they exist, you are three seconds from the answer; if they do not, the formula is still there.

  2. 2

    Read the form the question hands you

    Vertex form gives you the vertex without any work. Factored form gives you the roots. Standard form gives you the y-intercept. A question that supplies one of these usually wants the fact that form makes free.

  3. 3

    Use the discriminant when the question counts solutions

    You do not need to solve a quadratic to say how many real solutions it has. The sign of b^2 - 4ac decides it: positive means two, zero means one, negative means none.

  4. 4

    For rational expressions, factor everything first

    A quotient of polynomials almost always simplifies. Factor the numerator and the denominator, cancel the shared factor, and note the value that factor excludes. The excluded value is frequently what the question is really testing.

Worked examples

Each one is worked to the point where the next move is obvious, not to the answer alone.

1

What are the solutions to x^2 - 3x - 10 = 0?

  1. 1.Look for two numbers that multiply to -10 and add to -3. They are -5 and 2.
  2. 2.Write the factored form: (x - 5)(x + 2) = 0.
  3. 3.A product is zero when either factor is zero.

x = 5 and x = -2

2

The function f(x) = (x - 4)^2 - 9 has its minimum at which point?

  1. 1.The expression is already in vertex form, y = a(x - h)^2 + k, with h = 4 and k = -9.
  2. 2.Because the squared term is never negative and a is positive, the smallest value of f occurs when the square is zero.
  3. 3.That happens at x = 4, where f(4) = -9.

(4, -9)

3

For x not equal to 5, simplify (x^2 - 25) / (x - 5).

  1. 1.Recognise the numerator as a difference of squares: x^2 - 25 = (x - 5)(x + 5).
  2. 2.The quotient becomes (x - 5)(x + 5) / (x - 5).
  3. 3.Cancel the factor x - 5, which the question has already told you is nonzero.

x + 5

4

Write x^2 + 8x + 5 in the form (x + a)^2 + b.

  1. 1.Halve the coefficient of x to find a: 8 / 2 = 4.
  2. 2.(x + 4)^2 expands to x^2 + 8x + 16, which is 11 more than the original constant.
  3. 3.Subtract that surplus: (x + 4)^2 - 11.

(x + 4)^2 - 11

5

Solve 5^(2x) = 125.

  1. 1.Write both sides as powers of the same base: 125 = 5^3.
  2. 2.With equal bases the exponents must match: 2x = 3.
  3. 3.Divide by 2.

x = 1.5

6

Solve 1/x + 1/6 = 1/2 for x, where x is not zero.

  1. 1.Isolate the term containing x: 1/x = 1/2 - 1/6.
  2. 2.Use a common denominator: 1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3.
  3. 3.If 1/x = 1/3 then x = 3. Check: 1/3 + 1/6 = 1/2.

x = 3

Where marks are lost

Each of these has a matching wrong answer waiting among the four choices.

Giving one root when the question asks for both

A factored quadratic has two solutions. Questions often list one of them as a choice specifically to catch a reader who stopped after the first factor.

Adding exponents when the bases differ

The rule x^a times x^b = x^(a+b) needs the same base. It does not apply to 2^3 times 3^2, and a choice built on that mistake is usually present.

Dropping the excluded value

Cancelling x - 3 from a fraction is only valid where x is not 3. When a question specifies "for x not equal to 3", that condition is the point, not decoration.

Misreading the sign inside vertex form

In y = (x - 4)^2 + 1 the vertex is at x = 4, not x = -4. The form subtracts h, so the sign you read is the opposite of the sign you see.

Treating f(x + 1) as f(x) + 1

A function applied to a changed input is not the output changed by the same amount. For f(x) = x^2 + 1, f(x + 1) is x^2 + 2x + 2, not x^2 + 2. Substitute the whole new input, brackets and all.

Extrapolating a fitted model

A quadratic fitted from prices between 6 and 22 pounds will still return a number at 3 pounds, and that number carries almost no weight. When a question names the range the model came from, it is telling you where the answer stops being evidence.

Work these questions twice: once to get the answer, and once to ask which recognition would have saved you the algebra. The second pass is what makes this area fast rather than merely correct.

Now work the questions

112 Advanced Math questions with a worked explanation for each, on the SAT Math page.

Open SAT Advanced Math practice

The other SAT Math areas

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The arithmetic in this area is the easiest on the test. The difficulty is entirely in deciding what to compute. A question will give you five numbers and ask for a relationship between two of them, and the wrong pair is always available as a choice.

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SAT Geometry & Trigonometry

This is the smallest of the four Math areas and the most visual. A question that looks impossible in words is often obvious once drawn, and a sketch takes ten seconds. The formulas you need for area, volume and circles are supplied on the test; what is not supplied is the decision about which one applies.

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SAT® Advanced Math: Method and Practice | IQ Test Center