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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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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.

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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.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.

Every day at midnight. You get a fresh 12 easy, 12 medium and 12 hard.

No. Peakscor is an independent SAT practice tool, not affiliated with the College Board. Scores shown are estimates.

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Equivalent expressions · Lesson 3

Matching coefficients

Some questions say an equation is true **for all values of xx**. That means the two sides are the same expression written two ways, so each power of xx has the same coefficient on both sides.

The method: expand and simplify each side, line up the x2x^2-terms, the xx-terms and the constants, and set each pair of coefficients equal. In (x+4)(x−6)=x2+bx+c(x + 4)(x - 6) = x^2 + bx + c, the left side is x2−2x−24x^2 - 2x - 24, so b=−2b = -2 and c=−24c = -24.

Unknown constants can sit on either side. In (px+3)(x+q)=5x2+13x+6(px + 3)(x + q) = 5x^2 + 13x + 6, the x2x^2-terms give p=5p = 5 and the constants give 3q=63q = 6, so q=2q = 2. Check with the middle term: pq+3=10+3=13pq + 3 = 10 + 3 = 13, which matches.

Two shortcuts work because the equation holds for every xx. Substituting x=0x = 0 leaves only the constants, so it finds cc. Substituting x=1x = 1 turns ax2+bx+cax^2 + bx + c into a+b+ca + b + c, so it finds that sum in one step. Each substitution gives one equation, so when you need a single coefficient such as bb, expanding is usually the safer route.

Before you answer, reread which constant the question asks for. The values of the others are often among the choices.

Worked example Easy

(x+5)(x−3)=x2+bx+c(x + 5)(x - 3) = x^2 + bx + c

The given equation is true for all values of xx, where bb and cc are constants. What is the value of bb?

  1. −15-15
  2. −2-2
  3. 22Answer
  4. 88

How to solve it

  1. Expand the left side: x2−3x+5x−15=x2+2x−15x^2 - 3x + 5x - 15 = x^2 + 2x - 15.
  2. Match the xx-terms: b=2b = 2. (The constants give c=−15c = -15.)

Why each choice is right or wrong

  • A. Incorrect. −15-15 is cc, the constant term. The question asks for bb, the coefficient of xx.
  • B. Incorrect. This is a sign slip in the middle terms, −5x+3x=−2x-5x + 3x = -2x. The products are −3x-3x and +5x+5x, which give +2x+2x.
  • C. Correct. The middle products are −3x-3x and 5x5x, which combine to 2x2x, so b=2b = 2.
  • D. Incorrect. This adds 5 and 3 to get 88, ignoring the minus sign in x−3x - 3. The middle products are 5x5x and −3x-3x.

Worked example Medium

3(2x−1)2−4x=ax2+bx+c3(2x - 1)^2 - 4x = ax^2 + bx + c

The given equation is true for all values of xx, where aa, bb and cc are constants. What is the value of a+b+ca + b + c?

  1. −1-1Answer
  2. 55
  3. 77
  4. 1111

How to solve it

  1. Square first: (2x−1)2=4x2−4x+1(2x - 1)^2 = 4x^2 - 4x + 1.
  2. Multiply by 3: 12x2−12x+312x^2 - 12x + 3.
  3. Subtract 4x4x: 12x2−16x+312x^2 - 16x + 3. So a=12a = 12, b=−16b = -16, c=3c = 3, and a+b+c=−1a + b + c = -1.
  4. Faster: the equation is true for x=1x = 1, and the right side at x=1x = 1 is a+b+ca + b + c. The left side at x=1x = 1 is 3(1)2−4=−13(1)^2 - 4 = -1.

Why each choice is right or wrong

  • A. Correct. Expanding gives 12x2−16x+312x^2 - 16x + 3, and 12−16+3=−112 - 16 + 3 = -1. Substituting x=1x = 1 into the left side gives the same: 3(2−1)2−4=−13(2 - 1)^2 - 4 = -1.
  • B. Incorrect. This multiplies only the 4x24x^2 by 3, giving 12x2−4x+1−4x=12x2−8x+112x^2 - 4x + 1 - 4x = 12x^2 - 8x + 1, and 12−8+1=512 - 8 + 1 = 5. The 3 multiplies every term of the square.
  • C. Incorrect. This adds the 4x4x instead of subtracting it: 12x2−12x+4x+3=12x2−8x+312x^2 - 12x + 4x + 3 = 12x^2 - 8x + 3, and 12−8+3=712 - 8 + 3 = 7.
  • D. Incorrect. This squares term by term, writing (2x−1)2(2x - 1)^2 as 4x2+14x^2 + 1. Then 12x2+3−4x12x^2 + 3 - 4x gives 12−4+3=1112 - 4 + 3 = 11. The square has a middle term, −4x-4x.

Worked example Hard

(2x+p)(qx−5)=8x2+rx−15(2x + p)(qx - 5) = 8x^2 + rx - 15

The given equation is true for all values of xx, where pp, qq and rr are constants. What is the value of rr?

  1. −10-10
  2. 22Answer
  3. 1212
  4. 2222

How to solve it

  1. Expand the left side: 2qx2−10x+pqx−5p2qx^2 - 10x + pqx - 5p, which is 2qx2+(pq−10)x−5p2qx^2 + (pq - 10)x - 5p.
  2. Match the x2x^2-terms: 2q=82q = 8, so q=4q = 4.
  3. Match the constants: −5p=−15-5p = -15, so p=3p = 3.
  4. Match the xx-terms: r=pq−10=12−10=2r = pq - 10 = 12 - 10 = 2.

Why each choice is right or wrong

  • A. Incorrect. −10x-10x is only the outer product, 2x⋅(−5)2x \cdot (-5). The inner product, p⋅qx=12xp \cdot qx = 12x, also belongs to the xx-term.
  • B. Correct. 2q=82q = 8 gives q=4q = 4 and −5p=−15-5p = -15 gives p=3p = 3. The xx-terms are −10x-10x and pqx=12xpqx = 12x, so r=2r = 2. Check: (2x+3)(4x−5)=8x2−10x+12x−15=8x2+2x−15(2x + 3)(4x - 5) = 8x^2 - 10x + 12x - 15 = 8x^2 + 2x - 15.
  • C. Incorrect. 12 is pqpq, the inner product alone. The outer product, 2x⋅(−5)=−10x2x \cdot (-5) = -10x, also belongs to the xx-term.
  • D. Incorrect. This treats the outer product as +10x+10x, dropping the minus in −5-5, so 10+12=2210 + 12 = 22. But 2x⋅(−5)=−10x2x \cdot (-5) = -10x.

Trap Answering with the wrong constant

These questions define two or three constants and ask for one of them, or for a sum. The others are often among the choices. Write each value down as you find it, then reread the last line of the question.

Trap Matching before simplifying

Collect like terms on each side before you match. In 3(2x−1)2−4x3(2x - 1)^2 - 4x, the −4x-4x joins the −12x-12x from the square, so b=−16b = -16. Matching the −4x-4x alone to bxbx gives b=−4b = -4, which is wrong.

Desmos When Desmos isn’t faster

Expanding and matching by hand takes less time than setting up sliders for every constant. Use Desmos to check your answer instead.

Once you have values, type the left side with your numbers as one y=y = line and the right side as another. If the equation is true for all xx, the two graphs are the same curve, so you see only one.

Desmos also evaluates plain arithmetic, which helps with sums such as a+b+ca + b + c. Type the left side with 1 in place of xx, such as 3(2⋅1−1)2−4⋅13(2 \cdot 1 - 1)^2 - 4 \cdot 1, and Desmos shows the value, −1-1.

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Each test mirrors the digital SAT: 98 questions across four modules, 2 hr 14 min on official timing.

A little harder than the real SAT. These tests can run slightly tougher than test day, which makes them good practice: if you can handle these, the real one should feel easier. No question appears in more than one test, Challenge Tests included.

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Same 98-question format, but every question is drawn from the toughest SAT material: Challenging-tier math throughout and the hardest reading.

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