SAT Math Β· Algebra

Quadratic
Equations

20 carefully crafted problems β€” from foundations to tricky traps. Master every concept tested on the SAT.

Progress
0 / 0 answered
Unit 1
The Discriminant β€” How Many Solutions?
The most fundamental tool. Master this first.
🧠
Quick Memory Point
Discriminant = \(b^2 - 4ac\)
for \(ax^2 + bx + c = 0\)
D > 0 β†’ TWO real roots D = 0 β†’ ONE real root (repeated) D < 0 β†’ NO real roots KEY WORD: "distinct" = strictly MORE than one
1
Discriminant Β· Zero Solutions
\(5x^2 + 10x + 16 = 0\)
Easy
How many distinct real solutions does the equation above have?
discriminant = bΒ² βˆ’ 4ac β†’ check sign
ν•΄μ„€ (Korean Explanation)
νŒλ³„μ‹ D = bΒ² βˆ’ 4acλ₯Ό κ³„μ‚°ν•©λ‹ˆλ‹€.
μ—¬κΈ°μ„œ a = 5, b = 10, c = 16μ΄λ―€λ‘œ:
D = 10Β² βˆ’ 4(5)(16) = 100 βˆ’ 320 = βˆ’220
D < 0μ΄λ―€λ‘œ μ‹€μˆ˜ ν•΄κ°€ μ—†μŠ΅λ‹ˆλ‹€. μ΄μ°¨λ°©μ •μ‹μ˜ μ‹€μˆ˜ ν•΄μ˜ κ°œμˆ˜λŠ” νŒλ³„μ‹μ˜ λΆ€ν˜Έλ§ŒμœΌλ‘œ κ²°μ •λ©λ‹ˆλ‹€.
2
Discriminant Β· One Solution
\(x^2 - 6x + 9 = 0\)
Easy
How many distinct real solutions does this equation have?
perfect square β†’ D = 0 β†’ exactly ONE
ν•΄μ„€ (Korean Explanation)
이 식은 μ™„μ „μ œκ³±μ‹μž…λ‹ˆλ‹€: \((x-3)^2 = 0\)
x = 3 (쀑근, repeated root)
D = (βˆ’6)Β² βˆ’ 4(1)(9) = 36 βˆ’ 36 = 0
D = 0이면 ν•΄κ°€ μ •ν™•νžˆ 1κ°œμž…λ‹ˆλ‹€. 주의: "쀑근"이라도 "distinct solution"은 1개!
3
Discriminant Β· Two Solutions
\(2x^2 - 7x + 3 = 0\)
Easy
How many distinct real solutions does the equation have?
D = bΒ² βˆ’ 4ac β†’ positive = two roots
ν•΄μ„€ (Korean Explanation)
a = 2, b = βˆ’7, c = 3
D = (βˆ’7)Β² βˆ’ 4(2)(3) = 49 βˆ’ 24 = 25 > 0
D > 0μ΄λ―€λ‘œ μ„œλ‘œ λ‹€λ₯Έ μ‹€μˆ˜ ν•΄κ°€ 2κ°œμž…λ‹ˆλ‹€.
μΈμˆ˜λΆ„ν•΄: (2x βˆ’ 1)(x βˆ’ 3) = 0 β†’ x = 1/2, x = 3
Unit 2
Vertex Form & Completing the Square
Transform any quadratic to reveal its secrets.
πŸ“
Quick Memory Point
Vertex Form: \(y = a(x-h)^2 + k\)
Vertex = (h, k) Β· Axis of Symmetry: x = h
a > 0 β†’ opens UP, min at k a < 0 β†’ opens DOWN, max at k vertex x = βˆ’b/2a (standard form) TRAP: sign of h is flipped! y=(xβˆ’3)Β² β†’ h=+3
4
Vertex Form Β· Finding Vertex
\(y = (x - 4)^2 + 7\)
Easy
What is the vertex of the parabola defined by the equation above?
vertex form y = a(xβˆ’h)Β² + k β†’ vertex (h, k)
ν•΄μ„€ (Korean Explanation)
\(y = (x-4)^2 + 7\)μ—μ„œ h = 4, k = 7
꼭짓점은 (4, 7)μž…λ‹ˆλ‹€.
⚠️ ν”ν•œ μ‹€μˆ˜: (x βˆ’ 4)λ₯Ό 보고 h = βˆ’4둜 착각! λΆ€ν˜Έκ°€ λ°˜λŒ€λ‘œ λ“€μ–΄κ°€ μžˆμ–΄μš”. (x βˆ’ h)μ—μ„œ h의 λΆ€ν˜Έλ₯Ό κ·ΈλŒ€λ‘œ 읽어야 ν•©λ‹ˆλ‹€.
5
Vertex Form Β· Minimum Value
\(f(x) = 3(x + 2)^2 - 5\)
Easy
What is the minimum value of the function \(f(x)\)?
a > 0 β†’ opens up β†’ min = k value
ν•΄μ„€ (Korean Explanation)
a = 3 > 0μ΄λ―€λ‘œ 포물선은 μœ„λ‘œ μ—΄λ¦½λ‹ˆλ‹€ β†’ μ΅œμ†Ÿκ°’μ΄ μ‘΄μž¬ν•©λ‹ˆλ‹€.
꼭짓점 = (βˆ’2, βˆ’5) Β· μ΅œμ†Ÿκ°’ = k = βˆ’5
\((x+2)^2 \geq 0\)μ΄λ―€λ‘œ \(3(x+2)^2 \geq 0\), λ”°λΌμ„œ \(f(x) \geq -5\)
6
Standard β†’ Vertex Form Β· Tricky Sign
\(y = x^2 + 8x + 10\)
Medium
Which of the following correctly expresses the equation in vertex form?
complete the square: (b/2)Β² inside, subtract outside
ν•΄μ„€ (Korean Explanation)
μ™„μ „μ œκ³±μ‹ λ§Œλ“€κΈ°:
xΒ² + 8x + 10
= xΒ² + 8x + 16 βˆ’ 16 + 10   β† (8/2)Β² = 16 λ”ν•˜κ³  λΉΌκΈ°
= (x + 4)Β² βˆ’ 6
⚠️ ν”ν•œ μ‹€μˆ˜ 1: 16을 λ”ν•˜κ³  λΉΌλŠ” 것을 잊고 μƒμˆ˜ν•­λ§Œ λƒ…λ‘λŠ” 것 β†’ 10 κ·ΈλŒ€λ‘œ β†’ μ˜€λ‹΅
⚠️ ν”ν•œ μ‹€μˆ˜ 2: x + 4λ₯Ό 보고 꼭짓점 x = 4둜 착각 β†’ μ‹€μ œ 꼭짓점 x = βˆ’4
Unit 3
Factoring & Finding Roots
Work backwards from solutions to equations β€” and forwards.
πŸ”‘
Quick Memory Point
If roots are r and s:  \(a(x-r)(x-s) = 0\)
Sum = r + s = βˆ’b/a  Β·  Product = rs = c/a
sum = βˆ’b/a product = c/a zero product property: if AB=0 then A=0 or B=0 TRAP: x(xβˆ’3)=0 β†’ x=0 OR x=3 (don't forget x=0!)
7
Factoring Β· Zero Product Property
\(x^2 - 5x = 0\)
Easy
What are the solutions to the equation above? (Many students miss one.)
factor out x first β†’ x = 0 is a valid root!
ν•΄μ„€ (Korean Explanation)
x(x βˆ’ 5) = 0
영인수 λΆ„ν•΄: x = 0 λ˜λŠ” x βˆ’ 5 = 0
λ”°λΌμ„œ x = 0 λ˜λŠ” x = 5
⚠️ κ°€μž₯ ν”ν•œ μ‹€μˆ˜: 양변을 x둜 λ‚˜λˆ„λ©΄ x = 0 ν•΄λ₯Ό μžƒμ–΄λ²„λ¦½λ‹ˆλ‹€! μ ˆλŒ€ λ³€μˆ˜λ‘œ λ‚˜λˆ„μ§€ λ§ˆμ„Έμš”.
8
Sum & Product of Roots
\(x^2 - 7x + k = 0\)
Medium
If one solution of the equation is \(x = 3\), what is the value of \(k\)?
sum of roots = 7 β†’ find other root β†’ product = k
ν•΄μ„€ (Korean Explanation)
두 근의 ν•© = βˆ’(βˆ’7)/1 = 7
ν•œ 근이 3μ΄λ―€λ‘œ λ‹€λ₯Έ κ·Ό = 7 βˆ’ 3 = 4
두 근의 κ³± = k/1 = 3 Γ— 4 = 12
κ²€μ‚°: xΒ² βˆ’ 7x + 12 = (xβˆ’3)(xβˆ’4) = 0 βœ“
9
Roots β†’ Equation (Tricky)
Roots: \(x = \frac{1}{2}\) and \(x = -3\)
Medium
Which equation has the roots given above with integer coefficients?
fractional root β†’ multiply factor by denominator
ν•΄μ„€ (Korean Explanation)
근이 1/2μ΄λ―€λ‘œ μΈμˆ˜λŠ” (x βˆ’ 1/2) = 0, 즉 (2x βˆ’ 1) = 0
근이 βˆ’3μ΄λ―€λ‘œ μΈμˆ˜λŠ” (x + 3) = 0
(2x βˆ’ 1)(x + 3) = 2xΒ² + 6x βˆ’ x βˆ’ 3 = 2xΒ² + 5x βˆ’ 3 = 0
⚠️ D 함정: λΆ€ν˜Έλ₯Ό λ°”κΎΌ 2xΒ² βˆ’ 5x βˆ’ 3은 근이 βˆ’1/2κ³Ό 3μž…λ‹ˆλ‹€!
Unit 4
Quadratic Formula & Special Cases
When factoring fails β€” the universal solver.
⚑
Quick Memory Point
\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
memorize: "negative b plus/minus..." denominator = 2a (not just a!) simplify √D first before dividing TRAP: βˆ’b means flip the sign of b
10
Quadratic Formula Β· Application
\(x^2 + 4x + 1 = 0\)
Medium
What is the sum of all real solutions?
sum of roots = βˆ’b/a (no need for full formula!)
ν•΄μ„€ (Korean Explanation)
두 근의 ν•© = βˆ’b/a = βˆ’4/1 = βˆ’4
μ‹€μ œ 두 κ·Ό: x = (βˆ’4 Β± √12)/2 = βˆ’2 Β± √3
ν•©: (βˆ’2 + √3) + (βˆ’2 βˆ’ √3) = βˆ’4 βœ“
κΏ€νŒ: 두 근의 합을 ꡬ할 λ•Œ μ΄μ°¨λ°©μ •μ‹μ˜ 근의 곡식 전체λ₯Ό 계산할 ν•„μš”κ°€ μ—†μŠ΅λ‹ˆλ‹€!
11
Quadratic Formula Β· Exact Value
\(3x^2 - 2x - 2 = 0\)
Medium
Which expression represents the solutions?
plug into formula carefully: βˆ’b = βˆ’(βˆ’2) = +2
ν•΄μ„€ (Korean Explanation)
a=3, b=βˆ’2, c=βˆ’2
x = [2 ± √(4 + 24)] / 6 = [2 ± √28] / 6
√28 = 2√7μ΄λ―€λ‘œ: = [2 Β± 2√7] / 6 = (1 Β± √7) / 3
⚠️ A번: 약뢄을 μ•ˆ ν•œ 경우 β€” μˆ˜ν•™μ μœΌλ‘œ λ§žμ§€λ§Œ SATμ—μ„œλŠ” μ΅œμ’… 닡을 λ°˜λ“œμ‹œ μ•½λΆ„ν•΄μ•Ό ν•©λ‹ˆλ‹€!
Unit 5
Systems & Intersections
Where lines meet parabolas β€” a SAT favorite.
βœ‚οΈ
Quick Memory Point
Set equal β†’ rearrange β†’ use discriminant
tangent (1 intersection) β†’ D = 0 2 intersections β†’ D > 0 no intersection β†’ D < 0 substitute y = mx+b into y = axΒ²+bx+c
12
Line & Parabola Β· Number of Intersections
\(y = x^2 + 3\)   and   \(y = 2x\)
Medium
How many points of intersection do the two equations above have?
set equal β†’ check discriminant of combined equation
ν•΄μ„€ (Korean Explanation)
xΒ² + 3 = 2x
xΒ² βˆ’ 2x + 3 = 0
D = (βˆ’2)Β² βˆ’ 4(1)(3) = 4 βˆ’ 12 = βˆ’8 < 0
D < 0 β†’ ꡐ점이 μ—†μŠ΅λ‹ˆλ‹€. 직선이 포물선에 닿지 μ•Šμ•„μš”.
13
Tangent Line Β· Finding k (Hard Trap)
\(y = x^2 + kx + 4\)   and   \(y = 0\)
Hard
For what positive value of \(k\) does the parabola just touch the x-axis (tangent)?
tangent to x-axis = exactly 1 root = D must equal 0
ν•΄μ„€ (Korean Explanation)
x좕에 μ ‘ν•˜λ €λ©΄ D = 0이어야 ν•©λ‹ˆλ‹€.
D = kΒ² βˆ’ 4(1)(4) = 0
kΒ² = 16
k = Β±4
μ–‘μˆ˜ kλ₯Ό κ΅¬ν•˜λ―€λ‘œ k = 4
⚠️ 함정: k = βˆ’4도 μˆ˜ν•™μ μœΌλ‘œ λ§žμ§€λ§Œ λ¬Έμ œμ—μ„œ "positive value"λ₯Ό 묻고 μžˆμŠ΅λ‹ˆλ‹€!
Unit 6
Word Problems & Modeling
Apply quadratics to real-world scenarios.
🌍
Quick Memory Point
Projectile: \(h(t) = -16t^2 + v_0 t + h_0\) (feet) or \(-4.9t^2\) (meters)
max height β†’ vertex (axis: t = βˆ’b/2a) hits ground β†’ h(t) = 0 β†’ solve for t > 0 area problems β†’ set up equation, solve for dimension TRAP: reject negative solutions for physical lengths/time
14
Projectile Motion Β· Maximum Height
\(h(t) = -16t^2 + 64t + 5\)
Medium
A ball is thrown upward. Its height in feet after \(t\) seconds is given above. What is the maximum height reached?
max height = h(t) at vertex β†’ t = βˆ’b/2a
ν•΄μ„€ (Korean Explanation)
κΌ­μ§“μ μ˜ t μ’Œν‘œ: t = βˆ’b/2a = βˆ’64/(2Γ—βˆ’16) = βˆ’64/βˆ’32 = 2초
h(2) = βˆ’16(4) + 64(2) + 5 = βˆ’64 + 128 + 5 = 69ν”ΌνŠΈ
C 함정: 5λŠ” 초기 높이(t=0일 λ•Œ)μž…λ‹ˆλ‹€. μ΅œλŒ€ 높이가 μ•„λ‹™λ‹ˆλ‹€.
15
Area Word Problem Β· Setting Up Equation
Rectangle: length = width + 3, Area = 40
Medium
A rectangle's length is 3 more than its width. Its area is 40 square units. What is the width?
let width = x β†’ length = x+3 β†’ x(x+3) = 40
ν•΄μ„€ (Korean Explanation)
λ„ˆλΉ„ = x둜 λ†“μœΌλ©΄: x(x + 3) = 40
xΒ² + 3x βˆ’ 40 = 0
(x + 8)(x βˆ’ 5) = 0
x = βˆ’8 λ˜λŠ” x = 5
λ„ˆλΉ„λŠ” μ–‘μˆ˜μ—¬μ•Ό ν•˜λ―€λ‘œ x = 5
길이 = 5 + 3 = 8, κ²€μ‚°: 5 Γ— 8 = 40 βœ“
Unit 7
Advanced β€” Parametric & Tricky SAT Traps
The questions that separate 700 from 800.
🎯
Quick Memory Point
"Always true" β†’ analyze as function of parameter
substituting a root β†’ expression = 0 if both roots positive: sum > 0 AND product > 0 even degree β†’ both ends same direction TRAP: "no real solutions" β‰  "no solutions" (complex exist)
16
Parameter Β· Discriminant Condition
\(x^2 + mx + 9 = 0\)
Hard
For the equation above to have two distinct real solutions, which condition must hold?
D > 0 for two distinct roots β†’ solve for m
ν•΄μ„€ (Korean Explanation)
두 μ‹€μˆ˜ ν•΄λ₯Ό 가지렀면 D > 0
D = mΒ² βˆ’ 4(1)(9) > 0
mΒ² > 36
|m| > 6
m > 6 λ˜λŠ” m < βˆ’6
B 함정: m = Β±6이면 D = 0 β†’ 쀑근(ν•΄κ°€ 1개)! "distinct(μ„œλ‘œ λ‹€λ₯Έ)"μ΄λ―€λ‘œ BλŠ” μ˜€λ‹΅.
17
Hidden Quadratic Β· Substitution
\(x^4 - 5x^2 + 4 = 0\)
Hard
How many real solutions does the equation above have? (Hint: let \(u = x^2\))
biquadratic β†’ substitute u = xΒ² β†’ solve quadratic in u
ν•΄μ„€ (Korean Explanation)
u = x²둜 μΉ˜ν™˜ν•˜λ©΄: uΒ² βˆ’ 5u + 4 = 0
(u βˆ’ 1)(u βˆ’ 4) = 0 β†’ u = 1 λ˜λŠ” u = 4
u = 1 β†’ xΒ² = 1 β†’ x = Β±1 (2개)
u = 4 β†’ xΒ² = 4 β†’ x = Β±2 (2개)
총 μ‹€μˆ˜ ν•΄: 4개
핡심: uκ°€ μ–‘μˆ˜μ΄λ©΄ x = ±√u둜 2개의 μ‹€μˆ˜ ν•΄κ°€ λ‚˜μ˜΅λ‹ˆλ‹€.
18
Graph Interpretation Β· Leading Coefficient
\(f(x) = -2x^2 + 8x - 6\)
Medium
Which of the following is true about the graph of \(f(x)\)?
a < 0 β†’ opens down β†’ has MAXIMUM (not min)
ν•΄μ„€ (Korean Explanation)
a = βˆ’2 < 0 β†’ 포물선은 μ•„λž˜λ‘œ μ—΄λ¦Ό β†’ μ΅œλŒ“κ°’ 쑴재
꼭짓점 x = βˆ’b/2a = βˆ’8/(2Γ—βˆ’2) = 2
f(2) = βˆ’2(4) + 8(2) βˆ’ 6 = βˆ’8 + 16 βˆ’ 6 = 2
μ΅œλŒ“κ°’ = 2
19
Coefficient Identification Β· Tricky Rearrangement
\(3 - 2x = x^2\)
Medium
What is the value of the discriminant for the equation above?
rearrange to axΒ²+bx+c=0 form FIRST, then compute D
ν•΄μ„€ (Korean Explanation)
λ¨Όμ € ν‘œμ€€ν˜•μœΌλ‘œ 정리: 3 βˆ’ 2x = xΒ² β†’ xΒ² + 2x βˆ’ 3 = 0
a = 1, b = 2, c = βˆ’3
D = 2Β² βˆ’ 4(1)(βˆ’3) = 4 + 12 = 16
⚠️ ν”ν•œ μ‹€μˆ˜: μ›λž˜ 식 κ·ΈλŒ€λ‘œ 3 βˆ’ 2xμ—μ„œ b = βˆ’2, c = 3으둜 잘λͺ» 읽기! λ°˜λ“œμ‹œ axΒ²+bx+c=0 ν˜•νƒœλ‘œ λ¨Όμ € λ³€ν™˜ν•˜μ„Έμš”.
20
SAT Style Β· Both Conditions
\(kx^2 - 4x + k = 0\)
Hard
For which value of \(k\) does the equation have exactly one real solution?
D = 0 for exactly one root AND k β‰  0 (must stay quadratic!)
ν•΄μ„€ (Korean Explanation)
μ •ν™•νžˆ 1개의 μ‹€μˆ˜ ν•΄ β†’ D = 0
D = (βˆ’4)Β² βˆ’ 4(k)(k) = 0
16 βˆ’ 4kΒ² = 0
kΒ² = 4
k = Β±2
μ„ νƒμ§€μ—λŠ” k = 2κ°€ μžˆμŠ΅λ‹ˆλ‹€.
⚠️ A 함정 (k=0): k = 0이면 방정식이 βˆ’4x = 0이 λ˜μ–΄ 일차방정식이 λ©λ‹ˆλ‹€. 이차방정식이 μ•„λ‹ˆλ―€λ‘œ "이차방정식이 1개의 ν•΄λ₯Ό 가진닀"λŠ” 쑰건에 λ§žμ§€ μ•ŠμŠ΅λ‹ˆλ‹€!
πŸ“
Quadratic Equations Β· SAT Math Practice
20 problems Β· 7 units Β· all answers verified