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A quick run with three hearts. String right answers together and the questions get harder and the points multiply.
Expression of IdeasTransitions

Jazz pianist Mary Lou Williams composed hundreds of works. ______ she wrote arrangements for bands led by Duke Ellington and Benny Goodman.

AHowever,Your pick

BIn addition,Correct

Why: the second sentence adds another accomplishment. Nothing is being contrasted.

2
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Every question is tagged by domain and skill, and every answer comes with an explanation.
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9 easy, 9 medium, 9 hard and 9 challenging questions free every day, each with a full explanation. Filter by section, domain, skill and difficulty. Resets at midnight Central Time.

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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

C~~100%~~The trap

D104%

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

Turn each change into a multiplier.
Up 20% means × 1.20. Down 20% means × 0.80.
Apply them in order.
1.20 × 0.80 = 0.96
Read the answer.
The final price is 96% of the original. That’s B.
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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Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

Reading & Writing · Module 1
27 questions32 min
Reading & Writing · Module 2
27 questions32 min

Adapts to how you did on Module 1.
Break
10 min
Math · Module 1
22 questions35 min
Math · Module 2
22 questions35 min

Adapts to how you did on Module 1.

98 questions · 2 hr 14 min of testing, plus the break

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The Daily, plus 36 practice questions every day: 9 easy, 9 medium, 9 hard and 9 challenging, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too, as are the 20-question diagnostic, each Learn module’s overview and first lesson, and PEAK Survivor’s first map. 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.

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

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Nonlinear equations and systems · Lesson 5

Systems with a nonlinear equation

A nonlinear system has at least one equation that isn't a line, most often a parabola and a line. A solution is an ordered pair $(x, y)$ that makes both equations true, which is a point where the two graphs meet.

Substitute. When both equations start with $y =$ , set the right sides equal. For $y = x^2 - 3$ and $y = 2x$ : $x^2 - 3 = 2x$ , so $x^2 - 2x - 3 = 0$ , $(x - 3)(x + 1) = 0$ , and $x = 3$ or $x = -1$ .

**Find $y$ too.** Put each $x$ into the simpler equation: $y = 2(3) = 6$ and $y = 2(-1) = -2$ . The solutions are $(3, 6)$ and $(-1, -2)$ .

Count with the discriminant. After substituting, the quadratic in $x$ has as many real solutions as the graphs have intersection points. Discriminant positive: the line crosses the parabola twice. Zero: the line just touches it once. Negative: they never meet. For *exactly one solution, what is $k$ ?*, set this discriminant to 0.

Reading a graph. When the graphs are given, each intersection point is a solution. A point that is on only one of the graphs, such as the parabola's vertex or the line's $y$ -intercept, is not a solution of the system.

The graphs of

y = x^2 - 3

and

y = 2x

An upward-opening parabola with its vertex at (0, -3) and a line through the origin with slope 2. The line crosses the parabola at exactly two points, (-1, -2) and (3, 6). These are the two solutions of the system.

Worked example Easy

$y = x^2$

$y = x + 6$

Which ordered pair $(x, y)$ is a solution to the given system of equations?

$(-3, 9)$
$(0, 6)$
$(2, 4)$
$(3, 9)$ Answer

How to solve it

Both equations give $y$ , so set them equal: $x^2 = x + 6$ .
Standard form: $x^2 - x - 6 = 0$ , so $(x - 3)(x + 2) = 0$ and $x = 3$ or $x = -2$ .
Find $y$ : when $x = 3$ , $y = 9$ ; when $x = -2$ , $y = 4$ . The solutions are $(3, 9)$ and $(-2, 4)$ .
Only $(3, 9)$ is among the choices. Check it in both equations: $9 = 3^2$ and $9 = 3 + 6$ .

Why each choice is right or wrong

A. Incorrect. This reads the solution straight off the factor $x - 3$ with its sign kept, which gives $x = -3$ . The point is on the parabola, since $(-3)^2 = 9$ , but not on the line: $-3 + 6 = 3$ , not 9.
B. Incorrect. $(0, 6)$ is the line's $y$ -intercept. It is on the line but not on the parabola, since $0^2 = 0$ .
C. Incorrect. This reads the solution straight off the factor $x + 2$ with its sign kept, which gives $x = 2$ . The point is on the parabola, since $2^2 = 4$ , but not on the line: $2 + 6 = 8$ , not 4.
D. Correct. $x^2 = x + 6$ gives $x = 3$ or $x = -2$ , and $(3, 9)$ satisfies both equations: $3^2 = 9$ and $3 + 6 = 9$ .

Worked example Medium

$y = 2x^2 - 5x + 1$

$y = x - 3$

How many solutions does the given system of equations have?

Zero
Exactly one
Exactly twoAnswer
Infinitely many

How to solve it

Set the right sides equal: $2x^2 - 5x + 1 = x - 3$ .
Subtract $x$ and add 3: $2x^2 - 6x + 4 = 0$ . Divide by 2: $x^2 - 3x + 2 = 0$ .
Discriminant: $9 - 8 = 1$ , which is positive, so there are two solutions. (Factoring confirms it: $(x - 1)(x - 2) = 0$ , giving the points $(1, -2)$ and $(2, -1)$ .)

Why each choice is right or wrong

A. Incorrect. This adds $x$ instead of subtracting it, which gives $2x^2 - 4x + 4 = 0$ with discriminant $16 - 32 = -16$ . Subtracting $x$ gives $2x^2 - 6x + 4 = 0$ .
B. Incorrect. This reads the discriminant's value, 1, as the number of solutions. A positive discriminant means two solutions, whatever its size.
C. Correct. Substituting gives $x^2 - 3x + 2 = 0$ , whose discriminant is 1, so there are two solutions: $(1, -2)$ and $(2, -1)$ .
D. Incorrect. A line and a parabola can meet at most twice. Infinitely many solutions would need the two equations to describe the same graph.

Worked example Hard

$y = x^2 - 6x + k$

$y = 2x - 3$

In the given system, $k$ is a constant. The system has exactly one solution. What is the value of $k$ ?

$6$
$13$ Answer
$16$
$19$

How to solve it

Set the right sides equal: $x^2 - 6x + k = 2x - 3$ .
Move everything to the left: subtract $2x$ and add 3, so $x^2 - 8x + (k + 3) = 0$ .
Exactly one solution means the discriminant is 0: $64 - 4(k + 3) = 0$ , so $k + 3 = 16$ and $k = 13$ .
Check: with $k = 13$ , the equation is $x^2 - 8x + 16 = (x - 4)^2 = 0$ , so the line touches the parabola only at $x = 4$ .

Why each choice is right or wrong

A. Incorrect. This computes the discriminant before moving $2x$ , using $b = -6$ : $36 - 4(k + 3) = 0$ gives $k = 6$ . After moving $2x$ , the $x$ -coefficient is $-8$ .
B. Correct. Substituting gives $x^2 - 8x + (k + 3) = 0$ . A discriminant of 0 needs $k + 3 = 16$ , so $k = 13$ .
C. Incorrect. 16 is the value of $k + 3$ , the whole constant term. Subtract 3 to get $k$ .
D. Incorrect. This subtracts 3 instead of adding it when moving $-3$ to the left, which gives $k - 3 = 16$ and $k = 19$ . Moving $-3$ across means adding 3.

Trap A point on only one graph

Choices often include points that satisfy one equation but not the other: the vertex, an intercept, or a solution with the wrong sign. Before you pick an ordered pair, substitute it into both equations.

Trap Stopping at x

Substitution gives the $x$ -coordinates first. If the question asks for $y$ , an ordered pair or a sum such as $x + y$ , finish by putting each $x$ into one of the equations.

Desmos When Desmos is faster

Type both equations exactly as given, such as $y = x^2 - 3$ and $y = 2x$ . Click each point where the graphs cross to see its coordinates. Count the crossings to answer how many solutions.

For a constant such as $k$ , add the slider Desmos offers and type each answer choice as the slider's value. Watch for the value where the line just touches the parabola.

A line that nearly touches a parabola can look tangent when it isn't. If the picture is close, check with the discriminant of the combined equation.

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Full-length tests

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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Study the classic SAT vocabulary list. Browse, flip flashcards, or quiz yourself.

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Level up your SAT skills.

The real test, rehearsed.

One run a day. One skill at a time.

Play the Daily.

Miss one, learn why.

Drill that skill.

Beat your best.

Built around the real SAT blueprint.

Free daily practice

Real SAT timing

Adaptive full-length tests

Vocabulary

Your progress

Games

Every answer, explained.

Turn each change into a multiplier.

Apply them in order.

Read the answer.

Why not 100%?

What test day looks like.

Reading & Writing · Module 1

Reading & Writing · Module 2

Break

Math · Module 1

Math · Module 2

Peakscor by the numbers

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Peak

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What’s free?

What do Pro and Peak cost?

What does a full-length test include?

Do the tests adapt to me?

Which timing options are there?

How does the Daily work?

When do my free questions reset?

Is Peakscor affiliated with the College Board?

Your next point starts today.

Systems with a nonlinear equation

Worked example Easy

How to solve it

Why each choice is right or wrong

Worked example Medium

How to solve it

Why each choice is right or wrong

Worked example Hard

How to solve it

Why each choice is right or wrong

Trap A point on only one graph

Trap Stopping at x

Desmos When Desmos is faster

Your progress

Practice

Full-length tests

Challenge Tests Hardest of the hardest

Vocab

Free every day. Pro when you're serious.

FREE

PRO

PEAK