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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.
3
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Every question is tagged by section, domain, skill and difficulty, so you can practice exactly what you need. Full tests run like test day: two Reading & Writing modules, a break, then two Math modules.

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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 2

The discriminant and number of solutions

The discriminant is the part of the quadratic formula under the square root: $b^2 - 4ac$ , for $ax^2 + bx + c = 0$ . Its sign tells you how many real solutions there are without solving.

Positive: two real solutions. Zero: exactly one real solution, because $\pm 0$ gives the same value twice. Negative: no real solutions, because no real number squares to a negative.

On a graph, the solutions are the $x$ -intercepts of $y = ax^2 + bx + c$ . Two solutions means the parabola crosses the $x$ -axis twice. One solution means its vertex sits exactly on the $x$ -axis. No real solutions means it never reaches the $x$ -axis.

Always get standard form first. In $x^2 + 25 = 10x$ , the $10x$ belongs on the left: $x^2 - 10x + 25 = 0$ , so $b = -10$ and the discriminant is $100 - 100 = 0$ . Exactly one solution, $x = 5$ .

For *exactly one solution, what is $k$ ?*, set the discriminant equal to 0 and solve for $k$ . For no real solutions, solve discriminant $< 0$ . For two real solutions, solve discriminant $> 0$ .

Sum and product. When $ax^2 + bx + c = 0$ has two solutions, their sum is $-\frac{b}{a}$ and their product is $\frac{c}{a}$ . If a question asks only for the sum or product, use these and skip solving. For $2x^2 - 7x + 3 = 0$ , the sum is $\frac{7}{2}$ and the product is $\frac{3}{2}$ (the solutions are 3 and $\frac{1}{2}$ ).

y = x^2 - 4x + c

for

c = 3

,

c = 4

and

c = 6

Three upward-opening parabolas, each with its lowest point at x = 2. The lowest one (c = 3) crosses the x-axis twice, at (1, 0) and (3, 0). The middle one (c = 4) touches the x-axis only at its vertex, (2, 0). The highest one (c = 6) has its vertex at (2, 2) and never reaches the x-axis.

Worked example Easy

x^2 + 49 = 14x

How many distinct real solutions does the given equation have?

Zero
Exactly oneAnswer
Exactly two
Infinitely many

How to solve it

Standard form first: subtract $14x$ from both sides, $x^2 - 14x + 49 = 0$ .
$a = 1$ , $b = -14$ , $c = 49$ . Discriminant: $(-14)^2 - 4(1)(49) = 196 - 196 = 0$ .
A discriminant of 0 means exactly one real solution. Indeed $x^2 - 14x + 49 = (x - 7)^2$ , so $x = 7$ .

Why each choice is right or wrong

A. Incorrect. This computes the discriminant before moving $14x$ , using $b = 0$ and $c = 49$ : $0 - 196 < 0$ . Rewrite in standard form first, so $b = -14$ .
B. Correct. In standard form, $x^2 - 14x + 49 = 0$ , and $196 - 196 = 0$ . The equation is $(x - 7)^2 = 0$ , with the single solution $x = 7$ .
C. Incorrect. This adds $4ac$ instead of subtracting it: $196 + 196 > 0$ . The discriminant is $b^2 - 4ac$ , which is 0 here.
D. Incorrect. A quadratic equation has at most two solutions. Infinitely many would need the two sides to be identical.

Worked example Medium

2x^2 - 12x + k = 0

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

$-18$
$6$
$18$ Answer
$36$

How to solve it

Exactly one solution means the discriminant is 0.
$a = 2$ , $b = -12$ , $c = k$ : $(-12)^2 - 4(2)(k) = 144 - 8k$ .
$144 - 8k = 0$ , so $k = 18$ .
Check: $2x^2 - 12x + 18 = 2(x - 3)^2$ , which is 0 only at $x = 3$ .

Why each choice is right or wrong

A. Incorrect. This adds $4ac$ instead of subtracting it: $144 + 8k = 0$ . With $k = -18$ , the discriminant is $144 + 144 = 288$ , so there are two solutions.
B. Incorrect. This takes half of 12 and stops. Dividing by 2 gives $x^2 - 6x + \frac{k}{2} = 0$ , a perfect square only when $\frac{k}{2} = 9$ , the square of half of 6. With $k = 6$ , the discriminant is $144 - 48 = 96$ , so there are two solutions.
C. Correct. Setting the discriminant to 0 gives $144 - 8k = 0$ , so $k = 18$ , and the equation becomes $2(x - 3)^2 = 0$ .
D. Incorrect. This leaves out $a = 2$ : $144 - 4k = 0$ gives $k = 36$ . The discriminant is $b^2 - 4ac$ , and here $a = 2$ .

Worked example Hard

3x^2 - 21x + 5 = 0

What is the sum of the solutions to the given equation?

$-7$
$\frac{5}{3}$
$7$ Answer
$21$

How to solve it

Check that there are two solutions: $(-21)^2 - 4(3)(5) = 441 - 60 = 381 > 0$ .
The sum of the solutions is $-\frac{b}{a} = -\frac{-21}{3} = 7$ .
Solving with the formula would give $\frac{21 \pm \sqrt{381}}{6}$ . Adding them, the square roots cancel and leave $\frac{42}{6} = 7$ , the same answer with much more work.

Why each choice is right or wrong

A. Incorrect. This uses $\frac{b}{a}$ and forgets the negative sign: $\frac{-21}{3} = -7$ . The sum is $-\frac{b}{a}$ .
B. Incorrect. $\frac{5}{3}$ is $\frac{c}{a}$ , the product of the solutions, not the sum.
C. Correct. The sum of the solutions is $-\frac{b}{a} = \frac{21}{3} = 7$ , and the discriminant, 381, is positive, so there are two solutions to add.
D. Incorrect. 21 is $-b$ without dividing by $a = 3$ . The sum is $-\frac{b}{a} = \frac{21}{3} = 7$ .

Trap Using the discriminant before standard form

$b$ and $c$ only mean something when every term is on one side and 0 is on the other. In $x^2 + 49 = 14x$ , $b$ is $-14$ , not 0. Move everything to one side, then read $a$ , $b$ and $c$ with their signs.

Trap Mixing up one solution and no solution

Zero is the boundary. A discriminant of exactly 0 means one solution; a negative discriminant means none. In a no real solutions question, the value of $k$ that makes the discriminant 0 is often a choice, and it gives one solution, not zero.

Desmos When Desmos is faster

To count solutions, graph the equation with everything on one side, such as $y = x^2 - 14x + 49$ , and count the $x$ -intercepts. You can also graph each side as its own $y =$ and count the crossings.

For a constant such as $k$ , type $y = 2x^2 - 12x + k$ . Because $k$ isn't defined, Desmos offers to add a slider. Add it, click the slider's value and type each answer choice. Look for the choice that makes the vertex sit on the $x$ -axis (one solution) or lifts the parabola off it (none).

A vertex that sits just above or just below the axis can look like it touches. If the picture is close, compute $b^2 - 4ac$ by hand. For a sum or product question, $-\frac{b}{a}$ or $\frac{c}{a}$ is faster than any graph.

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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.

Challenge Tests Hardest of the hardest

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

Free every day. Pro when you’re serious.

Free

Pro

Peak

Questions, answered.

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.

The discriminant and number of solutions

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 Using the discriminant before standard form

Trap Mixing up one solution and no solution

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