SAT Advanced Algebra: Every Question Type and Method

The digital SAT's Math section has four content domains, and Advanced Math is one of the two biggest: about 35% of the section, or 13 to 15 of its 40 scored questions. The College Board splits it into three skills, and this course teaches the first two:

• Equivalent expressions: rewrite, factor and simplify polynomials, powers, roots and fractions.
• Nonlinear equations in one variable and systems of equations in two variables: quadratics, higher-degree, absolute value, exponential, radical and rational equations, and a line and a curve solved together.
• Nonlinear functions: graphs and models of quadratic, exponential and other functions, in the SAT Nonlinear Functions course.

Together, the two courses cover the Algebra 2 the SAT tests.

This guide covers every advanced algebra question type in six groups. The table below gives each one's method, and each section after it teaches the method, says when Desmos is faster, and gives you an SAT-style question to try. On most of these questions, the points are lost after the math is done: a root that fails the original equation, the wrong one of two roots, or the value of a stand-in letter instead of $x$.

How to answer an SAT advanced algebra question

The same four moves work on every question in this guide:

1. Read the last line first. Does it want an equivalent expression, a value of $x$, one of several roots, a count, or a constant like $k$?
2. Note every condition: $x>0$, $x\ne3$, positive, greatest, distinct.
3. Pick the faster route. When the factors are in view or the answer must be exact, work by hand. When the choices are long expressions in $x$, or the roots can be matched as decimals, Desmos is faster: it's built into the digital SAT for the whole Math section.
4. Find every solution, check it in the original, then choose.

Each section below turns these moves into the steps for its question type. The Desmos course teaches the calculator itself.

If these questions are new to you, start with lesson 1.

Expression questions never ask for a value of $x$. They ask which expression is equivalent to another, what a coefficient is after you rewrite, or which product a polynomial factors into. Two expressions are equivalent when they give the same value for every $x$, so one shared point never proves a match: the whole expression has to agree.

How to answer expression questions

1. Read what it wants: a matching form, a coefficient or a constant like $a+b+c$, or a factor.
2. Pick the route. With four long choices in one variable, graph the original and swap each choice onto a second line in Desmos: the right one sits exactly on top across the whole window. See equivalent expressions with Desmos. With a short rewrite, more than one variable, a letter like $k$, or a form you have to write yourself, work by hand.
3. Rewrite one layer at a time: distribute to every term, take out the greatest common factor first, and match coefficients power by power.
4. Check: expand your product, or test a second coefficient, before you choose.

Distribute and combine

A factor outside parentheses multiplies every term inside, and a minus sign in front works like $-1$, so it flips every sign. Then combine only like terms, with exactly the same variable part. Write every product out before you combine, so a middle term can't go missing. See simplify algebraic expressions SAT.

Factor

Factoring runs distribution backward. Take out the greatest common factor, then look again at what's left: a difference of squares, $A^2-B^2=(A-B)(A+B)$, a trinomial whose two numbers multiply to $ac$ and add to $b$, or four terms you can group in pairs. When an expression like $x-1$ repeats, call it $u$, factor, then put it back. See factor algebraic expressions SAT.

Identities and unknown coefficients

“For all values of $x$” means the two sides are the same polynomial, so match coefficients: $x^2$ with $x^2$, $x$ with $x$, constants with constants, with a $0$ for any missing power. A factor $x-c$ means the polynomial is $0$ at $x=c$, and a coefficient sum $a+b+c$ is the polynomial's value at $1$. See polynomial identities SAT.

Mistakes to avoid with equivalent expressions

The minus sign reaches every term: $-(x^2-7x-2)=-x^2+7x+2$. $3x^2$ and $4x$ have different powers, so $3x^2+4x$ is already simplified. And $x+3$ is $0$ at $x=-3$, so that's the input to plug in.

These questions give an expression full of powers, roots or fractions with $x$ in the denominator, and ask for an equivalent form. Most add a condition, like $x>0$ or $x\ne2$, and some ask only for the values $x$ can't take. The safe way through is to write every power with one base, and keep every restriction the original has.

How to answer exponent, radical and fraction questions

1. Read the conditions: $x>0$, the excluded values, and the form the answer takes.
2. Pick the route. With long choices in one variable, start with full-curve overlap in Desmos. With a short rewrite, more than one variable, or a question about excluded values, work by hand: a graph shows a missing value only as a hole you can't see. The Desmos domain restrictions lesson shows why.
3. Rewrite exactly: an outside exponent reaches every factor, the number in front included; power over root; and in a fraction, factor the top and bottom and cancel only whole factors.
4. Keep the restrictions: every value that made an original denominator $0$ stays excluded.

Exponent rules

With the same base, add exponents to multiply, subtract to divide, and multiply for a power of a power. A zero exponent gives $1$, and a negative exponent means a reciprocal: $x^{-2}=\frac{1}{x^2}$. Rewrite different numbers with one base before you combine: $4^2=(2^2)^2=2^4$. See SAT exponent rules.

Radicals and rational exponents

In $x^{m/n}$, the bottom number is the root and the top is the power, so $x^{3/4}=\sqrt[4]{x^3}$. To simplify a radical, pull out the perfect-power factor and leave the rest inside: $\sqrt{72x^5}=6x^2\sqrt{2x}$ for $x>0$. By hand, turn every form into exponents so you can compare them. See SAT radicals and rational exponents.

Rational expressions

Write the excluded values first, from every original denominator. Then factor and cancel factors, never terms. To add or subtract, build the least common denominator and multiply each top by what its bottom was missing. Often you'll use both routes: the graph picks the formula, and the original denominators give the exclusions. See simplify rational expressions SAT.

Mistakes to avoid with exponents and radicals

Rewrite $4^2$ as $2^4$ first, and $2^3\cdot2^4=2^7$. In $x^{2/3}$, the $3$ is the root and the $2$ the power: $\sqrt[3]{x^2}$. And the two $x$'s in $\frac{x+4}{x}$ are terms, not factors: at $x=2$ the fraction is $3$, not $4$.

Quadratic questions ask for the solutions, the one that's positive or greater, a number built from a root, how many real solutions there are, or a coefficient that makes something true. Factoring, graphing, the quadratic formula, the discriminant and the coefficients each answer a different one, so read the last line first: it picks the tool.

How to answer quadratic questions

1. Get $0$ on one side, in the form $ax^2+bx+c=0$, keeping every sign.
2. Read the last line and pick the tool. When the choices can be told apart as decimals, graph and click every $x$-intercept: see solve SAT equations with Desmos. When you can see the factors, factor. For an exact root with a square root, use the quadratic formula. For how many solutions, or the constant that gives one or none, use the discriminant. For a sum, a product or a coefficient, read it from the coefficients.
3. Keep every root, then apply the condition, like $x>0$ or the greater one, and do the last step the question asks for.

Factoring

If a product is $0$, one of its factors is $0$. That only works with $0$ on the other side, and the root of $x+6$ is $-6$, not $6$. Factor out $x$ instead of dividing by it, so $x=0$ stays a solution. See solve quadratic equations by factoring SAT.

The quadratic formula and the discriminant

Write $a$, $b$ and $c$ with their signs before you substitute, with negatives in parentheses. The discriminant, $b^2-4ac$, counts the real solutions: positive, zero or negative gives two, one or none. Complete the square when it makes one clean square. A graph's decimals can point to an exact answer, but they can't be one. See SAT quadratic formula and discriminant.

Sum and product of the solutions

For $ax^2+bx+c=0$, the solutions (the roots) add to $-\frac{b}{a}$ and multiply to $\frac{c}{a}$, so a sum, a product or a missing coefficient never needs the roots themselves. Known roots $r$ and $s$ give $a(x-r)(x-s)$, and a repeated root counts twice. See sum and product of roots.

Mistakes to avoid with quadratics

Move every term to one side before you set factors to $0$. The sum takes the minus sign: ${-\frac{-6}{2}=3}$. And a root $r$ gives the factor $x-r$, so roots $2$ and $-4$ give $(x-2)(x+4)$.

These equations look big but come apart: a product equal to $0$, a power of $x$ in every term, an expression that repeats, bars around one expression, or the same base in disguise, like $9$ and $27$. A cubic can have three roots and an absolute value equation two, and the question usually wants one of them, so find them all, then pick one.

How to answer absolute value, exponential and polynomial questions

1. Look for the structure: a product equal to $0$, a shared factor of $x$, a repeated expression to call $u$, an absolute value, or bases that are powers of one number.
2. Pick the route. When a short all-number equation asks for a root, graph it, or both of its sides, and read every crossing; widen the window until you've seen them all. See count equation solutions with Desmos. When the question wants a constant like $k$, letters in the answer, or proof you have every root, work by hand.
3. Split it: set each factor with $x$ in it to $0$; once the bars are alone, solve $\lvert A\rvert=d$ as $A=d$ and $A=-d$; or rewrite both sides with one base and set the exponents equal. After a substitution, put $x$ back.
4. Combine and choose: list every root, drop repeats when it says distinct, then apply the condition.

Higher-degree equations

Never divide by an expression with $x$ in it: you lose the root that makes it $0$. Factor instead, and split the product into branches. An equation with only even powers, like $x^4-13x^2+36=0$, is a quadratic in $u=x^2$, and each value of $u$ can give two values of $x$. See structured polynomial equations SAT.

Absolute value equations

Absolute value is distance from $0$, so get the bars alone first, then split. For $\lvert A\rvert=d$, a positive, zero or negative $d$ gives two, one or no solutions. When the other side has $x$ in it too, graph both original sides and click every intersection. See absolute value equations SAT.

Exponential equations

$8$, $16$, $32$ and $64$ are powers of $2$; $9$, $27$ and $81$ of $3$; $25$ and $125$ of $5$. Get the power alone, rewrite both sides with one base, then set the exponents equal. With no common base in sight, or a power that could give several solutions, graph both sides instead. See solve exponential equations SAT.

Mistakes to avoid with these equations

$x^3-4x=x(x-2)(x+2)$, so $x=0$ is a root too. The minus case belongs to the bars alone: get to $\lvert 2x+3\rvert=8$ first. And rewrite $9=3^2$ and $27=3^3$ before you set exponents equal: $2x+2=6x-3$.

Radical equations have $x$ under a square root; rational equations have $x$ in a denominator. Clearing either one by hand can create a value that solves the new equation but not the original, an extraneous solution. Formula questions ask you to solve for one letter in terms of the others. In all three, a value you find is only a candidate until it works in the original.

How to answer radical, rational and formula questions

1. Write the limits first: every value that makes an original denominator $0$, and for $\sqrt{A}=B$, $B$ can't be negative.
2. Pick the route. When the crossing gives the number or picks out a choice, graph the two original sides in Desmos and click where they meet: graphing the original can't add an extra answer. See solve SAT equations with Desmos. When one or two lines clear it, or the answer must be exact, work by hand, and always for a formula with letters.
3. Clear it: get the radical alone, then square both whole sides; multiply every term by the common denominator; in a formula, undo the outside first.
4. Check every candidate against the limits and in the original equation.

Radical equations

Squaring each term on its own isn't squaring the equation: get the radical alone, picture parentheses around each side, then square. And $\sqrt{25}$ is $5$, not $\pm5$: a square root names one number. See solve radical equations SAT.

Rational equations

Multiply every term by the least common denominator, constants included, then solve. A value that makes an original denominator $0$ is out, even when it solves the cleared equation. For only the sum or product of the solutions, read it from the coefficients once you've checked that no excluded value is a root. See solve rational equations SAT.

Rearranging formulas

Undo the outside first, doing each step to both whole sides. When the target shows up in two terms, gather them and factor it out. For $U^2=R$, write $U=\pm\sqrt R$ and let the condition pick the branch. Do these by hand: sample values can catch a slip but can't prove a formula. See rearrange nonlinear formulas SAT.

Mistakes to avoid with radical and rational equations

Check in the original: $\sqrt{2(-3)+15}=3$, not $-3$. Subtract $4$ first, so $\sqrt{3x+10}=7$ and $x=13$. And a square hides its sign: $y-6=\pm\sqrt R$, then let the condition choose.

These systems pair a line with a curve, like a parabola, a circle or $xy=12$, or two curves. A solution is a point that works in both equations, and a line can cross a parabola twice, touch it once or miss it. Questions ask for one coordinate, a value built from the point, how many solutions there are, or the constant that gives exactly one. Keep every point, then answer what's asked.

How to answer nonlinear system questions

1. Read what it asks: a coordinate, a value, a count or a constant, and any condition like $x>0$.
2. Pick the route. When a variable is already alone, or both equations start with $y=$, substitute. When a decimal is fine, the question only wants a count, or the graphs might meet several times, graph both and click every intersection: see solve systems of equations with Desmos. For a constant, let the graph show where the count changes and prove it with algebra.
3. Keep every solution: solve every factor, find the other coordinate for each, and check each point in both equations.
4. Answer the question: the coordinate, value or count it asks for.

Substitute and solve

Put the whole expression in: with $y=x-1$, $xy=12$ becomes $x(x-1)=12$. Then get $0$ on one side and solve the quadratic. Each root gives its own point. See linear nonlinear systems SAT.

How many solutions

Substitute to get one quadratic, then read its discriminant: positive, zero or negative means the graphs cross twice, touch once or miss. For $y=x^2+k$ and $y=2x$, $x^2-2x+k=0$ has one solution when $4-4k=0$, so $k=1$. Graphs that look tangent at one zoom level may not be, so prove the constant by hand.

Mistakes to avoid with nonlinear systems

Label each point's $x$ and $y$ before you answer. The whole expression goes in, in parentheses: $x(x-1)=12$. And a quadratic has two roots: $(x-4)(x+3)=0$ gives $x=4$ as well.