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Equivalent expressions: SAT Math notes
Rewriting an expression in a different form that is still equal to it: expanding, combining like terms, factoring, using exponent rules, and working with fractions that have x in them.
Study it with worked examples and 7 practice questionsPractise Equivalent expressions questions
What this topic tests
Two expressions are equivalent when they give the same value for every allowed value of x. In this skill you are not solving for x. You are changing how an expression looks without changing what it is worth. The SAT either asks you to pick the matching form from four choices, or asks for a missing constant that makes two forms match.
Almost every question uses a few basic moves: distribute and combine like terms, factor, apply the exponent rules, and combine or split fractions. Easy questions need one move. Hard ones chain two or three together, so the real skill is working neatly and watching your signs.
You also have a safety net. If you are unsure, pick a simple number for x, such as 2 or 3, and plug it into the original and into each choice. The equivalent one gives the same value. Desmos can run the same check: graph the original and a choice, and if the two curves sit exactly on top of each other, they are equivalent.
Key ideas
- Like terms have the same variable raised to the same power. You can add 3x2 and 5x2 to get 8x2, but 3x2 and 5x stay separate.
- A minus sign in front of brackets flips the sign of every term inside: -(a-b+c)=-a+b-c.
- (a+b)2=a2+2ab+b2, not a2+b2. Squaring a bracket always creates a middle term.
- Difference of squares: a2-b2=(a-b)(a+b).
- Exponent rules: xa·xb=xa+b, (xa)/(xb)=xa-b, (xa)b=xab, and n√(xm)=xm/n.
- If two expressions are equal for every x, their matching coefficients are equal: the x2 numbers match, the x numbers match, and the constants match.
- Testing a value like x=2 in the original and in each choice is a quick way to check your work or break a tie. Avoid 0 and 1, because many different expressions agree there.
The 4 kinds of Equivalent expressions question
Sorted from every Equivalent expressions question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Expand and combine like terms
How to spot it. The expression has brackets to multiply out, or polynomials being added or subtracted, and the choices are plain polynomials like ax2+bx+c. Harder versions ask for just one coefficient, or include an unknown constant such as p that you have to find.
How to answer it:
- Distribute everything. Multiply each term in one bracket by each term in the other, and push any minus sign through its whole bracket.
- Group like terms by power: all the x2 terms together, all the x terms together, all the plain numbers together.
- Add the coefficients in each group and write the result from the highest power down.
- If the question asks for one coefficient or an unknown constant, only work out the terms you need, set the matching coefficients equal, and solve.
- If you have time, check in Desmos: graph the original and your answer. If the curves overlap exactly, they are equivalent.
The trap. Sign slips are the number one problem. When you subtract a bracket, every term inside changes sign, not just the first one. The other favourite wrong answer comes from squaring a bracket as (a+b)2=a2+b2 and losing the middle term 2ab.
2. Factoring and rewriting a quadratic
How to spot it. The question asks for a factor, or for the expression written as a product such as (x+a)(x+b) or k(…), or in the form (x+h)2+k. Often a constant inside the new form is missing and you have to find it.
How to answer it:
- Pull out the greatest common factor first: any number or power of x that every term shares.
- Look for a special pattern: a difference of squares a2-b2=(a-b)(a+b), or a perfect square a2+2ab+b2=(a+b)2.
- Otherwise, for x2+bx+c, find two numbers that multiply to c and add to b. For the form (x+h)2+k, take h as half the x coefficient, then fix the constant.
- If the question gives a form with a missing constant, expand that form and match coefficients. The constant term is usually the quickest to match.
- Check by multiplying your factors back out. For harder ones, graph the expression in Desmos: an x-intercept at r means x-r is a factor.
The trap. Getting the signs backwards. A factor x-r goes with a zero at x=r, so if x=-9 makes the expression zero, the factor is x+9. Also be careful with choices that look like a perfect square, such as (x-√(5))2, when the expression has no middle term. That one is really the difference of squares (x-√(5))(x+√(5)).
3. Exponents and radicals
How to spot it. The expression has roots, fractional or negative exponents, or powers being multiplied and divided, and the choices show the same thing written another way.
How to answer it:
- Turn every root into a fractional exponent: n√(xm)=xm/n. A plain square root is the power 1/2.
- Handle the numbers separately from the variables. For example, √(16)=4.
- Combine powers of the same base: add exponents when multiplying, subtract when dividing, and multiply when raising a power to a power.
- Rewrite your result in the style of the choices. A negative exponent means the power belongs in the denominator: x-1=1/x.
- If the choices look very different from each other, test a value such as x=2 on your calculator or in Desmos for the original and each choice.
The trap. Mixing up the rules: multiplying exponents when you should add them, or adding when you should subtract. Say the rule as you use it: same base multiplied means add the exponents. And a root of a power divides the exponent; it does not subtract from it.
4. Rational expressions
How to spot it. There is a fraction with x in the denominator. You either combine two fractions into one, simplify a fraction, or split it into a whole part plus a remainder, like 4+37/(x-9).
How to answer it:
- To add or subtract fractions, get a common denominator: multiply the top and bottom of each fraction by the factor it is missing, then combine the numerators.
- To simplify, factor the top and the bottom completely, then cancel any factor that appears in both.
- To split a fraction into a whole part plus a remainder, write the numerator in terms of the denominator. For example, 4x+1=4(x-9)+37.
- If the question names constants like r and t, match coefficients in the final numerator.
- Check by testing an allowed value of x in the original and in your answer. They should give the same number.
The trap. Canceling terms instead of factors. In (x+7)/(x+2) you cannot cancel the x's, because they are added, not multiplied. Only cancel something that multiplies the whole top and the whole bottom. When subtracting fractions, the minus sign applies to the whole second numerator.
Practise Equivalent expressions
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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