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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

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Equivalent expressions · Lesson 1

Expanding and combining polynomials

Two expressions are equivalent when they give the same value for every value of xx the question allows: every real xx, unless the question limits xx, such as *for x>0x > 0*. On the SAT you usually show it by rewriting one expression until it looks like one of the choices.

Distribute to remove parentheses: a number or term outside multiplies every term inside, so 2x(x−7)=2x2−14x2x(x - 7) = 2x^2 - 14x. Then combine like terms, terms with the same variable raised to the same power. 4x24x^2 and −x2-x^2 combine; 6x26x^2 and 4x4x do not.

Subtracting a polynomial means subtracting every term in it. (2x2+x−1)−(x2−3x+4)(2x^2 + x - 1) - (x^2 - 3x + 4) becomes 2x2+x−1−x2+3x−42x^2 + x - 1 - x^2 + 3x - 4, which is x2+4x−5x^2 + 4x - 5. Think of the minus sign as −1-1 distributed through the parentheses.

To multiply two binomials, multiply each term of the first by each term of the second, four products in all: (x+5)(2x−1)=2x2−x+10x−5=2x2+9x−5(x + 5)(2x - 1) = 2x^2 - x + 10x - 5 = 2x^2 + 9x - 5.

Three special products save time: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 and (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2. Squaring a binomial always produces a middle term, 2ab2ab or −2ab-2ab.

Worked example Easy

Which expression is equivalent to (4x2−3x+5)−(x2+2x−7)(4x^2 - 3x + 5) - (x^2 + 2x - 7)?

  1. 3x2−x−23x^2 - x - 2
  2. 3x2−5x−23x^2 - 5x - 2
  3. 3x2−5x+123x^2 - 5x + 12Answer
  4. 5x2−x−25x^2 - x - 2

How to solve it

  1. Distribute the minus sign to every term of the second polynomial: 4x2−3x+5−x2−2x+74x^2 - 3x + 5 - x^2 - 2x + 7.
  2. Combine the x2x^2-terms: 4x2−x2=3x24x^2 - x^2 = 3x^2.
  3. Combine the xx-terms: −3x−2x=−5x-3x - 2x = -5x.
  4. Combine the constants: 5+7=125 + 7 = 12. The result is 3x2−5x+123x^2 - 5x + 12.

Why each choice is right or wrong

  • A. Incorrect. This subtracts only the x2x^2-term and adds the other two: 4x2−x2−3x+2x+5−7=3x2−x−24x^2 - x^2 - 3x + 2x + 5 - 7 = 3x^2 - x - 2. The minus sign applies to all three terms of the second polynomial.
  • B. Incorrect. This changes the signs of x2x^2 and 2x2x but not of −7-7, so the constant becomes 5−7=−25 - 7 = -2. Subtracting −7-7 adds 7, so the constant is 12.
  • C. Correct. Distributing the minus gives 4x2−3x+5−x2−2x+74x^2 - 3x + 5 - x^2 - 2x + 7, which combines to 3x2−5x+123x^2 - 5x + 12.
  • D. Incorrect. This adds the two polynomials instead of subtracting: 4x2+x2=5x24x^2 + x^2 = 5x^2, −3x+2x=−x-3x + 2x = -x and 5−7=−25 - 7 = -2.

Worked example Medium

Which expression is equivalent to (2x−3)2−(x+1)(x−1)(2x - 3)^2 - (x + 1)(x - 1)?

  1. 3x2+103x^2 + 10
  2. 3x2−12x+103x^2 - 12x + 10Answer
  3. 3x2−12x+83x^2 - 12x + 8
  4. 3x2−6x+103x^2 - 6x + 10

How to solve it

  1. Square the binomial with the pattern (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2: (2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9.
  2. Use the difference of squares: (x+1)(x−1)=x2−1(x + 1)(x - 1) = x^2 - 1.
  3. Subtract the whole product: 4x2−12x+9−(x2−1)=4x2−12x+9−x2+14x^2 - 12x + 9 - (x^2 - 1) = 4x^2 - 12x + 9 - x^2 + 1.
  4. Combine like terms: 3x2−12x+103x^2 - 12x + 10.

Why each choice is right or wrong

  • A. Incorrect. This squares each term separately, writing (2x−3)2(2x - 3)^2 as 4x2+94x^2 + 9, which gives 4x2+9−x2+1=3x2+104x^2 + 9 - x^2 + 1 = 3x^2 + 10. Squaring a binomial produces a middle term, here −12x-12x.
  • B. Correct. (2x−3)2=4x2−12x+9(2x - 3)^2 = 4x^2 - 12x + 9 and (x+1)(x−1)=x2−1(x + 1)(x - 1) = x^2 - 1. Subtracting gives 4x2−12x+9−x2+1=3x2−12x+104x^2 - 12x + 9 - x^2 + 1 = 3x^2 - 12x + 10.
  • C. Incorrect. This subtracts x2x^2 but not −1-1, writing 4x2−12x+9−x2−14x^2 - 12x + 9 - x^2 - 1. Subtracting x2−1x^2 - 1 means subtracting both terms, so the constant is 9+1=109 + 1 = 10.
  • D. Incorrect. This drops the 2 in the middle term 2ab2ab. The middle term is 2⋅2x⋅3=12x2 \cdot 2x \cdot 3 = 12x, not 6x6x, so it is −12x-12x.

Trap Squaring term by term

(x+5)2(x + 5)^2 is not x2+25x^2 + 25. Write it as (x+5)(x+5)(x + 5)(x + 5) and you get four products: x2+5x+5x+25=x2+10x+25x^2 + 5x + 5x + 25 = x^2 + 10x + 25. A choice with the middle term missing is a common trap.

Trap A minus sign that stops after one term

In A−(B+C)A - (B + C), the minus reaches CC too: A−B−CA - B - C. So −(x2−1)-(x^2 - 1) is −x2+1-x^2 + 1. The SAT often lists the answer you get when the sign change stops partway through the parentheses.

Desmos When Desmos is faster

Type the original expression as y=(2x−3)2−(x+1)(x−1)y = (2x - 3)^2 - (x + 1)(x - 1), then type each choice as its own y=y = line. Equivalent expressions have the same graph, so the right choice lies exactly on top of the original curve, and each wrong choice separates from it somewhere.

Click the colored circle beside an expression to hide or show its graph, which makes it easier to see which curve covers the original. Two different curves can look alike in one window, so zoom in or out before you decide.

For a short expression like 4x(x+2)−3x4x(x + 2) - 3x, expanding by hand is faster than typing five lines.

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