SAT MATH�Nonlinear Solving → updated

Post-session review + practice

Week of Jan 12, 2026

Table of contents

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02

Quadratics + Its forms

Discriminant

Vertex form

Factored form

Converting between standard + vertex forms.

It relates to # of (real) solutions

03

Shortcuts for Quads

04

05

Exponential growth, decay

“Percent of” problems

06

Exponents

We do quick rapid-fire drill. Know like back of your hand ☺

3 Forms of Quadratics

Standard form

Vertex form

Factored form

• Vertex is at (h, k)
• If |a| > 1 —> narrow graph
• |a| < 1 —> wider =
• How to translate between standard & vertex form: complete the square

NonLinear solving – vertex of quadratic

SAMPLE #1

Full graphed solutions here: https://www.desmos.com/calculator/q6vyysuxtt

Digital SAT: MATH

NonLinear solving – vertex of quadratic

SAMPLE #1

Digital SAT: MATH

NonLinear solving – vertex of quadratic

QUICK DRILLS

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https://www.numerade.com/courses/sat/sat-math-quadratics/vertex-form-of-a-quadratic-overview/

Digital SAT: MATH

NonLinear solving – advanced quadratics

Ex 1: Convert standard to Vertex Form

y = x² − 2x − 8

Step 1: Group x² and x terms: y = (x² − 2x) − 8

Step 2: Complete the square inside parentheses

(b/2)² = (−2/2)² = (−1)² = 1

Add and subtract 1: y = (x² − 2x + 1) − 8 − 1

Step 3: Factor the perfect square trinomial

y = (x − 1)² − 9

Step 4: Identify vertex from vertex form

Vertex Form: y = (x − 1)² − 9

Vertex: (1, −9)

Ex 2: Convert standard to Vertex Form, a ≠ 1

2x² + 12x + 17 = y

Factor out 2 from x terms: y = 2(x² + 6x) + 17

Complete the square: (6/2)² = 9

y = 2(x² + 6x + 9) + 17 − 2(9)

🡪 2(x + 3)² + 17 − 18 🡪 2(x + 3)² - 1

Vertex Form: y = 2(x + 3)² − 1

Vertex: (−3, −1)

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How is it transformed with respect to y = x² ?

Transformation: 2f(x + 3) − 1

Digital SAT: MATH

NonLinear solving – advanced quadratics

Completing the Square - Key Rule

To complete the square, divide the coefficient of the linear term (b) by 2, then SQUARE the result.

y = x² + bx + (b/2)² is a perfect square = (x + b/2)²

Digital SAT: MATH

Nonlinear Solving

MATH

NUMBER OF SOLUTIONS

Teacher note:

The Linear Solving chapter dealt with this topic for linear equations.

b^2 - 4ac < 0

b^2 - 4ac > 0

Digital SAT: MATH

Nonlinear Solving

MATH

Digital SAT: MATH

Digital SAT: MATH

Digital SAT: MATH

Nonlinear solving – mini review of quads

Digital SAT: MATH

Nonlinear solving – mini review of quads

Digital SAT: MATH

Nonlinear Solving – product and sum of x-roots

MATH

Digital SAT: MATH

Real SAT Problem – product and sum of x-roots

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Digital SAT: MATH

Real SAT Problem – product and sum of x-roots

Derivation below: 

Vieta's formulas tell you that for ax² + bx + c = 0:

• Sum of roots: r₁ + r₂ = -b/a
• Product of roots: r₁ × r₂ = c/a

These come from factoring. If a quadratic has roots r₁ and r₂, it can be written as:

a(x−r1​)(x−r2​)=0

Expanding this:

a[x² − (r₁​+r₂​)x  +  r₁​*r₂​]=0

Ax² − a(r₁​+r₂)x + a(r₁​*r₂)=0

Comparing to ax² + bx + c = 0:

• b = -a(r₁ + r₂), so r₁ + r₂ = -b/a

c = a(r₁r₂), so r₁r₂ = c/a

Digital SAT: MATH

Table of contents

01

02

Quadratics + Its forms

Discriminant

Vertex form

Factored form

Converting between standard + vertex forms.

It relates to # of (real) solutions

03

Shortcuts for Quads

04

05

Exponential growth, decay

“Percent of” problems

06

Exponents

We do quick rapid-fire drill. Know like back of your hand ☺

✔

✔

✔

Nonlinear Solving

MATH

GROWTH AND DECAY (MEMORIZE)

Teacher note:

Students need to memorize these. This is a high-yield topic of the math section. The SAT loves to trick you with these formats: Either they play around with the “# of changes” exponent part, or they ask you to “reverse engineer” and find the ORIGINAL amount, given final amount.

Digital SAT: MATH

Growth/Decay Easy Mode

MATH

Digital SAT: MATH

Growth/Decay Easy Mode

MATH

Teacher note:

Hidden Plug-Ins were covered a while ago, so this is a good reminder. The clue is that the question is about the relationship between variables.

Concept Checker: what does 3 represent?

In this case, it’s number of “cycles”, number

of periodic cycles. It’s not always number of months, years or days. See next problem

Digital SAT: MATH

Growth/Decay Medium Mode

MATH

Teacher note:

This question is more important than Q8 because it introduces the twist of changing units. Where # of cycles or changes isn’t equal to simple y-years.

Digital SAT: MATH

Digital SAT: MATH

Digital SAT: MATH

Real SAT Problems: Growth/Decay Practice Mode

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Digital SAT: MATH

Linear equations practice

Do at home according to email instructions please

Linear equations (practice at home independently)

Set 2: Linear Equations with Parameters

Key Strategy: Substitute intercept coordinates to create equations relating parameters.

Problem 2.1

Line m is defined by ax + by = 24. Line m passes through (6, 0) and (0, 4). What is the value of a + b?

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Problem 2.2

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The equation 3x + ky = 21 represents a line passing through the point (p, 0), where p is a positive constant. Which expression represents k in terms of p?

• k = 21 − 3p
• k = (21 − 3p)/p
• k = 7/p
• k cannot be determined from the given information

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Problem 2.3

• Line ℓ passes through (a, 0) and (0, b) where a and b are positive. The equation of line ℓ can be written as x/a + y/b = 1. If the line also passes through (2, 3) and a = 2b, find the values of a and b.

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Problem 2.4

• The line rx + sy = rs passes through points (s, 0) and (0, r), where r and s are positive constants with r ≠ s. If the line also passes through (2, 3), and s = r + 1, find r.

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Problem 2.5

• A line in the form kx + (k+2)y = 3k passes through the origin for exactly one value of k. What is this value of k?

Digital SAT: MATH

Extra practice (at home maybe)?

• Set 2 Answers
• 2.1: a = 4, b = 6, so a + b = 10
• 2.2: D — The x-intercept alone doesn't determine k; we need another point
• 2.3: a = 4, b = 2
• 2.4: r = 5 (so s = 6)
• 2.5: k = 0 — Only k = 0 makes the constant term 0, allowing the line through origin

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Digital SAT: MATH

Set 2 step-by-step solutions

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Problem 2.1

Line m is defined by ax + by = 24. Line m passes through (6, 0) and (0, 4). What is the value of a + b?

Solution:

Substitute each point into ax + by = 24:

Point (6, 0): a(6) + b(0) = 24 → a = 4

Point (0, 4): a(0) + b(4) = 24 → b = 6

Answer: a + b = 4 + 6 = 10�

Problem 2.2

The equation 3x + ky = 21 represents a line passing through the point (p, 0), where p is a positive constant. Which expression represents k in terms of p?

Solution:

Substitute (p, 0) into 3x + ky = 21:

3(p) + k(0) = 21

3p = 21

p = 7

Since the y-term disappears, we only learn that p = 7. The value of k has no effect on whether (p, 0) is on the line—any value of k works as long as p = 7.

Answer: k cannot be determined from the given information�

Problem 2.3

Line l passes through (a, 0) and (0, b) where a and b are positive. The equation of line l can be written as x/a + y/b = 1. If the line also passes through (2, 3) and a = 2b, find the values of a and b.

Solution:

Substitute (2, 3) into x/a + y/b = 1:

2/a + 3/b = 1

Since a = 2b, substitute:

2/(2b) + 3/b = 1

1/b + 3/b = 1

4/b = 1

b = 4, so a = 2(4) = 8

Answer: a = 8, b = 4

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Problem 2.4

The line rx + sy = rs passes through points (s, 0) and (0, r), where r and s are positive constants with r ≠ s. If the line also passes through (2, 3), and s = r + 1, find r.

Solution:

Substitute (2, 3) into rx + sy = rs:

r(2) + s(3) = rs

2r + 3s = rs

Since s = r + 1, substitute:

2r + 3(r + 1) = r(r + 1)

2r + 3r + 3 = r² + r

5r + 3 = r² + r

r² - 4r - 3 = 0

Use Quad formula (or you can try factoring, but I don’t think it works in this case to clean whole numbers) to solve. 

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Problem 2.5

A line in the form kx + (k+2)y = 3k passes through the origin for exactly one value of k. What is this value of k?

Solution:

Substitute (0, 0) into kx + (k+2)y = 3k:

k(0) + (k+2)(0) = 3k

0 = 3k

k = 0

Verification: When k = 0, the equation becomes 0x + 2y = 0, which simplifies to y = 0 (the x-axis), which does pass through the origin. ✓

Answer: k = 0

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Digital SAT: MATH

SAT Reading�"Logic Mapping” of passages + �sprinkling of variety ☺

LOGICAL INFERENCE

Correct Answer: (D)

Logic Mapping the passage. The study showed:

• Ship sounds alone: Decreased foraging
• Ship sounds + sonic pulses: Also decreased foraging
• Difference between these two conditions: is sonic sounds. But + sonic sounds had same result as without it. SOO… sonic sounds are Negligible  → sonic pulses along not contribute much to behavorial diff”s

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Why other choices are wrong:

• A: Says narwhals forage at different depths in the two conditions → contradicts "negligible differences"
• B: Discusses sound travel distance but doesn't explain why effects were similar
• C: About overall sensitivity, not about why the two exposure types produced similar results

READING NOV 2025, V1

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Digital SAT: MATH

READING

Correct Answer: (A) surreptitiously

Analysis: we can see why there’s a 50/50 between succinctly and surreptitously esp if you dont know what (A) means.

• Succinctly not quite fits because it’s not the direct opposite of “widely/openly available”. Succinct means efficient in speech or writing. Ex: That was a succinct summary.
• Surreptitiously = discreetly, secretly. Secretly is the best way to memorize it. 🙂�

HARD VOCAB

Digital SAT: MATH

DUAL TEXTS

Answer: (D) b/c text 2 states pattern on non-boreal research in text #1 is not generalized across diff ecosystems. 

READING

Digital SAT: MATH

DUAL TEXTS

Answer: (D) b/c text 2 states pattern on non-boreal research in text #1 is not generalized across diff ecosystems. 

Logic Mapping

In Text 1:

• Cites "previous research conducted in non-boreal ecosystems"
• This research found: warming benefits invasive species more than native species
• Text 1 implies this finding might explain Alaska's situation (invasive peashrub establishing as climate warms)

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Text 2:

• Actual study in boreal Alaska (Fairbanks)
• Tracked both native (O. secunda) and invasive (C. arborescens) species
• Finding: Despite temperature variation between years, neither species showed significant variation in growth patterns
• Conclusion: Warming did NOT differentially benefit the invasive over the native species

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Why D Captures This:

• Text 2's finding casts doubt on the generalizability of the non-boreal research mentioned in Text 1.
• The non-boreal research suggested a general pattern (warming → invasives benefit more), but 2 shows this pattern doesn't hold up in boreal Alaska. So Text 1’s research not generalization. 
• Why the other choices don't work:
• A: Text 2 doesn't describe methodology solutions
• B: Text 2 doesn't offer an alternative explanation; it shows the expected pattern didn't occur
• C: Text 2 doesn't challenge the validity of the non-boreal findings themselves; it shows they don't apply universally
• Bottom line: D correctly identifies that Text 2 shows the non-boreal research doesn't generalize to boreal ecosystems like Alaska.

READING

Digital SAT: MATH

LISTS AND ”:”

READING

Super nitty gritty grammar stuff:

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The underlined independent clauses are complete thoughts because they make sense by themselves because they have a subject verb, and object (except in the second sentence where no object is needed). A colon should only be used after an independent clause. AKA if you cannot put a period (.) after the clause, then you cannot put a colon.

Digital SAT: MATH

CLAIMS: WEAKEN

Answer: (B) 

What are researcher’s claim?

• Languages developed from the original African language 
• Each new language retained fewer sounds as humans spread farther from Africa
• This explains why Rarámuri (Mexico, far from Africa) has only 20 sounds while Taa (Africa) has 100+... so 

CORE CLAIM: Languages farther from Africa result in Fewer sounds

What is opposite of core claim?  Far away languages have LOTS Of sounds. 

Europe and W asia are pretty far from Africa, and “tend to have more sounds than languages that emerged in africa do.”

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Why others are wrong: 

A: More variety in Africa than South America actually supports the claim (Africa = origin = more diversity)

C: Central America having fewer sounds than Europe and being farther away doesn't address the Africa comparison

D: Consistency within Africa doesn't address the distance-based sound loss pattern�

READING

Digital SAT: MATH