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Digital SAT Prep · Digital SAT Reading, Writing & Math

Advanced Math

Nonlinear equations, quadratics, polynomials, exponentials and functions.

Seven concepts on the nonlinear toolkit the Digital SAT Advanced Math domain expects — quadratics, vertex form, the quadratic formula, Vieta's shortcuts, exponentials and function notation. Each card states the idea, works a real SAT-style problem, and flags the trap that costs marks.

  • Digital SAT Prep
  • Hard level
  • 7 concepts
  • 5 practice questions

1The discriminant counts real roots

The discriminant D = b^2 - 4ac tells you how many real roots a quadratic ax^2+bx+c=0 has before you solve it. When D>0 the parabola crosses the x-axis twice, when D=0 it just touches once, and when D<0 it never reaches the axis. On the SAT, exactly one solution is the signal to set D=0 and solve for the unknown parameter — you never need to find the root itself.

Figure. When c = 9 the parabola y = x^2 - 6x + 9 touches the x-axis at x = 3 and nowhere else — the graph of D = 0. Lift c above 9 and the whole curve sits above the axis; drop c below 9 and it crosses twice.

How it works

  1. Read the coefficientsWrite the equation as ax^2+bx+c=0 and identify a, b, and c, including signs.
  2. Compute DEvaluate D = b^2 - 4ac. A negative middle term still squares to a positive contribution.
  3. Match sign to rootsTwo roots if D>0, one repeated root if D=0, none if D<0. For a parameter, set D equal to the count you need.
What D predicts
Sign of DReal rootsGraph
D > 0two distinctcrosses the axis twice
D = 0one repeatedtouches the axis once
D < 0nonestays above or below the axis

Find c for exactly one root

For what value of c does x^2 - 6x + c = 0 have exactly one real solution?

  • One root means D = 0; here a=1, b=-6D = 36 - 4c
  • Set 36 - 4c = 04c = 36
  • Solve for cc = 9

Pro tip. Exactly one solution always means D=0, not that you should factor or use the quadratic formula. The repeated root is x = -b/(2a) = 3 here, but the SAT question only wanted c.

If x^2 - 6x + c = 0 has no real solutions, which must be true of c?
  1. c < 9
  2. c = 9
  3. c > 9
  4. c = 0

No real roots means D < 0, so 36 - 4c < 0, hence c > 9. At c = 9 the discriminant is zero and there is one repeated root.

2Factoring before the formula

When ax^2+bx+c has small integer coefficients, factoring is faster than the quadratic formula. Find two numbers that multiply to ac and add to b, split the middle term, then pull out a common factor from each pair. Each factor set to zero gives a root. If no clean pair exists after a quick scan, switch to the formula — do not grind through a messy factorisation under time pressure.

Figure. The graph crosses the axis at x=2 and x=3 — exactly the roots from (x-2)(x-3)=0. A parabola that factors over the integers always has x-intercepts at those integer roots.

How it works

  1. Target the productFor x^2+bx+c, hunt two integers whose product is c and sum is b. For ax^2+bx+c with a\neq 1, the product target is ac.
  2. Split and groupRewrite the middle term using your pair, group into two binomials, and factor each group.
  3. Zero-product ruleIf (x-r_1)(x-r_2)=0, then x=r_1 or x=r_2. Each bracket gives one root.

Solve by factoring

What are the solutions to x^2 - 5x + 6 = 0?

  • Need two numbers: product 6, sum -5-2 and -3
  • Factor as (x-2)(x-3)=0roots x=2 or x=3
  • Check: (2)^2-5(2)+6 and (3)^2-5(3)+60 in both cases

Pro tip. On the SAT, try factoring first whenever the constant term has only small factors. The question bank repeats the same product-sum pairs — 6 with sum 5, 6 with sum -5, 15 with sum -8 — so a ten-second scan often finishes the problem.

Which factorisation is equivalent to x^2 - 9 = 0?
  1. (x-3)(x-3)=0
  2. (x-3)(x+3)=0
  3. (x+9)(x-1)=0
  4. (x-9)(x+1)=0

x^2-9 is a difference of squares: (x-3)(x+3)=0, giving x=\pm 3. The repeated factor (x-3)^2 would yield only x=3, and the other pairs do not expand to x^2-9.

3Vertex form and the extreme value

Completing the square rewrites y=ax^2+bx+c as y=a(x-h)^2+k, exposing the vertex (h,k) directly. The axis of symmetry is the vertical line x=h, and a controls direction: a>0 gives a minimum at (h,k), a<0 a maximum. On the SAT, minimum or maximum value questions almost always want the output at the vertex, not the input that produces it — find h=-b/(2a) only when you need to evaluate f(h).

Figure. The lowest point sits at (4,-1) — below the axis because -1<0. Every minimum-value question on a upward-opening parabola asks for that y-coordinate.

How it works

  1. Locate the axisFor ax^2+bx+c, the axis of symmetry is x = -b/(2a). That x-value is h in vertex form.
  2. Evaluate at the vertexSubstitute x=h into the original expression. The output is k, the minimum or maximum.
  3. Read the sign of aa>0 means the parabola opens up and k is a minimum; a<0 means k is a maximum.

Minimum value of a quadratic

The function f is defined by f(x)=x^2-8x+15. What is the minimum value of f(x)?

  • Axis: x = -(-8)/(2\cdot 1)x = 4
  • Evaluate: f(4) = 4^2 - 8(4) + 1516 - 32 + 15 = -1
  • Completing the square check: (x-4)^2 - 1minimum -1 at x=4

Pro tip. The answer to a minimum-value question is the y-coordinate at the vertex, here -1, not the x-coordinate 4. If the options list only x-values, you still need to evaluate f at that x.

For g(x)=-2(x+1)^2+5, which statement is true?
  1. The vertex is (1,5) and 5 is a minimum
  2. The vertex is (-1,5) and 5 is a maximum
  3. The vertex is (-1,5) and 5 is a minimum
  4. The axis of symmetry is y=5

Vertex form a(x-h)^2+k gives vertex (h,k)=(-1,5). Here a=-2<0, so the parabola opens down and 5 is a maximum, not a minimum.

4The quadratic formula

When factoring fails, x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} always works for ax^2+bx+c=0 with a\neq 0. The \pm produces two roots when D>0, one when D=0, and complex roots when D<0 — though the Digital SAT sticks to real roots. Simplify the radical before dividing: factor out perfect squares so the final answer stays in lowest radical form.

Figure. When integer factoring fails, the quadratic formula is the reliable next move — same roots, different route.

How it works

  1. Standard formMove everything to one side so the equation reads ax^2+bx+c=0. Identify a, b, c with signs.
  2. Compute D and \sqrt{D}Evaluate D=b^2-4ac. Simplify the square root — e.g. \sqrt{12}=2\sqrt{3}.
  3. Substitute and reducePlug into x=\dfrac{-b\pm\sqrt{D}}{2a} and cancel any common factor in the fraction.

Irrational roots

Solve x^2 + 4x + 1 = 0.

  • a=1, b=4, c=1; D = 16 - 4D = 12
  • \sqrt{12} = 2\sqrt{3}; x = \dfrac{-4 \pm 2\sqrt{3}}{2}x = -2 \pm \sqrt{3}
  • Two distinct roots because D=12>0x = -2+\sqrt{3} or x = -2-\sqrt{3}

Pro tip. Always divide -b and \sqrt{D} by 2a together — here dividing top and bottom by 2 turns -4\pm 2\sqrt{3} into -2\pm\sqrt{3}. Skipping that cancellation is the most common algebra slip on radical answers.

How many real solutions does x^2 + x + 5 = 0 have?
  1. Two
  2. One
  3. None
  4. Infinitely many

D = 1 - 20 = -19 < 0, so there are no real solutions. The quadratic formula would involve \sqrt{-19}, which is not a real number.

5Sum and product without solving

For ax^2+bx+c=0 with roots r_1 and r_2, Vieta's relations give r_1+r_2=-b/a and r_1 r_2=c/a — no factoring or formula required. The SAT often asks for a sum or product directly, or for a new quadratic whose roots are a translate of the original. Expand (x-r_1)(x-r_2) and match coefficients when you need the equation back from its roots.

Figure. Vieta maps coefficients straight to sum and product. Reach for these when the stem asks for r1+r2 or r1 r2 without the roots themselves.

How it works

  1. Read a, b, cFrom ax^2+bx+c=0, identify the three coefficients including signs.
  2. Apply VietaSum =-b/a. Product =c/a. Divide by a even when a\neq 1.
  3. Build a new equation if neededGiven roots p and q, the monic quadratic is (x-p)(x-q)=0, i.e. x^2-(p+q)x+pq=0.
Vieta at a glance
QuantityFormulaFrom x^2-5x+6=0
Sum of roots-b/a5
Product of rootsc/a6
Monic equation from roots p,qx^2-(p+q)x+pq=0(x-2)(x-3)=0

Sum and product

For 2x^2 - 10x + 12 = 0, find the sum and product of the roots without solving.

  • a=2, b=-10, c=12coefficients identified
  • Sum = -(-10)/25
  • Product = 12/26

Pro tip. Divide by a first mentally: dividing the whole equation by 2 gives x^2-5x+6=0, where the sum 5 and product 6 are visible from the coefficients. The full Vieta formulas are the general version of that shortcut.

If the roots of x^2 + kx + 12 = 0 have product 12 and sum -7, what is k?
  1. k = 7
  2. k = -7
  3. k = 12
  4. k = -12

Vieta gives sum =-b/a=-k. Setting -k=-7 yields k=7. The product c/a=12 is already consistent. Answering k=-7 treats the sum as equal to k instead of -k.

6Exponential growth and decay

A quantity modeled by y=a\,b^x starts at y=a when x=0 and changes by the fixed factor b each time x increases by 1. Growth happens when b>1; decay when 0<b<1. In percent form, b=1+r for growth rate r and b=1-r for decay — subtract 1 from the base to read the percent. Doubling time and half-life questions reduce to finding when b^x equals 2 or \tfrac{1}{2}.

Figure. Each unit step to the right multiplies the height by 1.08 — that fixed ratio is what makes the curve exponential rather than linear. The starting height 500 is the coefficient a.

How it works

  1. Identify a and bMatch the equation to y=a\,b^x. The coefficient a is the initial value; the base b is the per-step multiplier.
  2. Classify growth or decayb>1 means growth; 0<b<1 means decay. Compare b to 1, not to 0.
  3. Convert to a percent rateWrite b=1+r. Then r as a percent is the per-period change — e.g. b=1.08 means 8% growth per period.

Reading the growth factor

A population is modeled by P = 500(1.08)^t, with t in years. What does 1.08 represent?

  • Form P = a\,b^t with a=500, b=1.08initial population 500
  • b = 1 + r gives r = 1.08 - 1r = 0.08
  • Interpret as a percent per year8% annual growth

Pro tip. In a\,b^t, subtract 1 from the base to read the percent rate: 1.08\Rightarrow 8\% growth. A base of 0.85 would mean 15% decay, because 0.85=1-0.15.

If 2^{x+3}=64, what is 3^x?
  1. 27
  2. 81
  3. 9
  4. 243

Since 64=2^6, match exponents to get x+3=6, so x=3. Then 3^x=3^3=27. The trap is stopping at x=3 without evaluating the expression the question actually asked for.

7Function notation and zeros

f(x) is the output when x is substituted into the rule — so f(3) means replace every x with 3 and simplify. The zeros of f are the x-values where f(x)=0, the same as the x-intercepts of its graph. Factored form f(x)=(x-r_1)(x-r_2) exposes zeros directly: set each factor to zero. For nested expressions like f(g(x)), evaluate the inner function first, then feed that output into f.

Figure. Function notation is substitution: feed the input into the rule and simplify. Zeros are the inputs that make the output zero — the x-intercepts of the graph.

How it works

  1. Evaluate f(a)Substitute x=a everywhere in the rule. Follow order of operations — exponents before multiplication.
  2. Find zerosSet f(x)=0 and solve. In factored form, each factor (x-r)=0 gives a zero r.
  3. Compose inside outFor f(g(x)), compute g(x) first, then apply f to that result.

Evaluate and locate zeros

If f(x)=2x^2-3, what is f(4), and what are the zeros of f?

  • f(4) = 2(4^2) - 32(16)-3 = 29
  • Set 2x^2-3=0x^2 = \tfrac{3}{2}
  • Take square rootsx = \pm\sqrt{\tfrac{3}{2}}

Pro tip. Squaring comes before multiplying by 2 in f(4): 2(4^2)-3, not (2\cdot 4)^2-3. For zeros of 2x^2-3, the graph crosses the axis symmetrically at \pm\sqrt{3/2}.

If g(x)=x^2+2x, what is g(-2)?
  1. 0
  2. 4
  3. -4
  4. 8

g(-2)=(-2)^2+2(-2)=4-4=0. The trap is forgetting that (-2)^2=4, not -4.

Notes

  • Quadratic Solutions: A quadratic ax^2+bx+c=0 is solved by factoring, completing the square, or the quadratic formula; the discriminant b^2-4ac tells how many real roots exist.
  • Vertex Form: y=a(x-h)^2+k has vertex (h,k); the axis of symmetry is x=h and a controls direction and width.
  • Exponential Growth/Decay: y = a\cdot b^x grows when b>1 and decays when 0<b<1; a is the initial value.
  • Function Notation and Roots: The zeros of f(x) are the x-intercepts; f(x)=(x-r_1)(x-r_2) shows roots directly as r_1 and r_2.

Formulas

  • Quadratic formula: x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Discriminant: D = b^2 - 4ac (two real roots if D>0, one if D=0, none if D<0)
  • Sum and product of roots: r_1+r_2 = -\dfrac{b}{a}, r_1 r_2 = \dfrac{c}{a}
  • Vertex of a parabola: x = -\dfrac{b}{2a}
  • Exponential model: y = a\,b^x with growth factor b = 1 + r

Exam traps & shortcuts

  • Use the discriminant to answer 'how many solutions' questions without solving the whole quadratic.
  • When a quadratic must have exactly one solution, set b^2 - 4ac = 0 and solve for the unknown parameter.
  • For root questions, recall r_1+r_2=-b/a and r_1 r_2=c/a to find sums or products without computing each root.

Reference tables

Every line here should be recoverable from the concepts above, not merely recalled.

Advanced Math formula sheet
ToolFormulaWatch for
Quadratic formulax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}Simplify \sqrt{D} before dividing by 2a
DiscriminantD = b^2 - 4acTwo / one / none real roots when D>0 / =0 / <0
Vertex xx = -\dfrac{b}{2a}Minimum-value questions want f(x) there, not x alone
Vertex formy = a(x-h)^2 + kVertex (h,k); axis x=h
Sum of rootsr_1+r_2 = -\dfrac{b}{a}Works without finding either root
Product of rootsr_1 r_2 = \dfrac{c}{a}Divide by a, not just c
Exponential modely = a\,b^xPercent rate: b=1+r; subtract 1 from b
Zeros from factorsf(x)=(x-r_1)(x-r_2)Zeros are r_1 and r_2 directly

Recap

Read only this the night before.

Discriminant
D=b^2-4ac counts roots before you solve. Exactly one solution means D=0 — solve for the parameter, not the root.
Factoring
Product-sum pair, then zero-product rule. No clean pair in ten seconds? Reach for the formula.
Vertex
x=-b/(2a) locates the extreme; the minimum or maximum value is f at that x, usually written as k in vertex form.
Formula
x=\dfrac{-b\pm\sqrt{D}}{2a} always works. Simplify the radical, then cancel with 2a.
Vieta
Sum =-b/a, product =c/a. The SAT asks for these without making you find the roots first.
Exponentials
y=a\,b^x: a is the start, b the multiplier. b=1.08 means 8% growth per step.
Functions
f(a) means substitute. Zeros are x-intercepts. Compose inside out: g first, then f.

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