1.4 Quadratic, Polynomial, Exponential & Logarithmic Equations

Ateneo College Entrance Test ACET

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1 Chapter 1: Mathematics

1.1 Real Numbers, Ratios, Rates & Percentages 1.2 Algebraic Expressions, Polynomials & Linear Equations 1.3 Functions, Domain/Range, Graphs & Transformations 1.4 Quadratic, Polynomial, Exponential & Logarithmic Equations 1.5 Plane Geometry: Triangles, Circles, Polygons & Perimeter/Area 1.6 Coordinate Geometry: Lines, Distance, Midpoints & Conic Sections 1.7 Trigonometry: Right Triangle Ratios, Unit Circle & Basic Identities 1.8 Arithmetic & Geometric Sequences, Series & Word Patterns 1.9 Permutations, Combinations, Probability & Statistics

2 Chapter 2: English Proficiency

2.1 Subject-Verb Agreement, Verb Tenses & Aspect 2.2 Pronoun-Antecedent Agreement, Case, Misplaced & Dangling Modifiers 2.3 Sentence Structure, Run-Ons, Fragments & Parallelism 2.4 Contextual Vocabulary, Word Roots, Prefixes & Suffixes 2.5 Advanced Synonyms, Antonyms & Nuanced Vocabulary 2.6 Verbal Analogies: Relationship Types & Solving Strategies 2.7 Punctuation, Capitalization, Mechanics & Common Errors 2.8 Diction, Idiomatic Expressions, Prepositions & Frequently Confused Words

3 Chapter 3: Reading Comprehension

3.1 Identifying Main Ideas, Central Themes & Paragraph Summaries 3.2 Locating Explicit Details, Supporting Evidence & Data 3.3 Drawing Logical Inferences & Unstated Conclusions 3.4 Analyzing Author's Purpose, Point of View, Tone & Attitude 3.5 Interpreting Figurative Language, Tone Words & Context Clues 3.6 Text Structure, Rhetorical Devices & Evaluating Arguments

4 Chapter 4: Logical and Abstract Reasoning

4.1 Number & Letter Series Patterns & Formulas 4.2 Abstract Reasoning: Matrix Patterns & Figure Rotations 4.3 Formal Syllogisms, Categorical Logic & Deductive Proofs 4.4 Coding-Decoding, Cipher Patterns & Symbol Operations 4.5 Spatial Visualization, Paper Folding & Venn Diagram Logic

5 Chapter 5: General Knowledge

5.1 Philippine History, Government, Constitution & Civics 5.2 Integrated Science: Biology, Chemistry, Physics & Earth Science 5.3 Philippine & World Geography, Basic Economics & Cultural Facts 5.4 Current Events, World Landmarks, Literature & General Trivia

6 Chapter 6: ACET Essay Writing & Exam Strategies

6.1 ACET Essay: Thesis Development, Structuring & Argumentation 6.2 Time Allocation, Pacing Strategies & Subtest Tactics 6.3 Guessing Strategies (No Penalty), Test Anxiety & Exam Day Readiness

Free exam prep ACET (Ateneo College Entrance Test) Practice Questions 108 questions Study Guide 35 topics Cheat Sheet 56 facts Flashcards 50 cards

1.4 Quadratic, Polynomial, Exponential & Logarithmic Equations

Key Takeaways Explain This Topic AI

Apply Vieta's formulas to calculate the sum and product of quadratic and polynomial roots without solving explicitly.

Utilize the discriminant Delta = b^2 - 4ac to classify quadratic root types and determine graphical x-intercept counts.

Exploit logarithmic identities to combine logarithmic expressions and solve exponential equations via common bases.

Model growth, decay, and compound interest using continuous and discrete exponential functions. Last updated: July 2026

Quadratic, Polynomial, Exponential & Logarithmic Equations Non-linear equations appear frequently in the ACET Mathematics subtest. Mastering polynomial root behaviors, quadratic optimizations, exponential growth curves, and logarithmic properties will allow you to solve complex equations quickly. 1. Quadratic Function & Equation Analysis General Form: a x 2 + b x + c = 0 ( a ≠ 0 ) ax^2 + bx + c = 0 \quad (a \neq 0) a x 2 + b x + c = 0 ( a  = 0 ) The Quadratic Formula & The Discriminant ( Δ \Delta Δ ) x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c Δ > 0 \Delta > 0 Δ > 0 and a perfect square: Two distinct, rational real roots. Δ > 0 \Delta > 0 Δ > 0 and non-square: Two distinct, irrational real roots. Δ = 0 \Delta = 0 Δ = 0 : One real repeated root (double root, parabola tangent to x-axis). Δ < 0 \Delta < 0 Δ < 0 : Two complex conjugate non-real roots (parabola does not touch x-axis). Vieta's Formulas (Quadratic & Higher Degree) For a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 with roots r 1 r_1 r 1 and r 2 r_2 r 2 : Sum of Roots: r 1 + r 2 = − b a \text{Sum of Roots: } r_1 + r_2 = -\frac{b}{a} Sum of Roots: r 1 + r 2 = − a b Product of Roots: r 1 r 2 = c a \text{Product of Roots: } r_1 r_2 = \frac{c}{a} Product of Roots: r 1 r 2 = a c Sum of Reciprocals: 1 r 1 + 1 r 2 = r 1 + r 2 r 1 r 2 = − b / a c / a = − b c \text{Sum of Reciprocals: } \frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{-b/a}{c/a} = -\frac{b}{c} Sum of Reciprocals: r 1 1 + r 2 1 = r 1 r 2 r 1 + r 2 = c / a − b / a = − c b Sum of Squares: r 1 2 + r 2 2 = ( r 1 + r 2 ) 2 − 2 r 1 r 2 = ( − b a ) 2 − 2 ( c a ) = b 2 − 2 a c a 2 \text{Sum of Squares: } r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2 r_1 r_2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2 - 2ac}{a^2} Sum of Squares: r 1 2 + r 2 2 = ( r 1 + r 2 ) 2 − 2 r 1 r 2 = ( − a b ) 2 − 2 ( a c ) = a 2 b 2 − 2 a c Vertex Form & Optimization Vertex Form: y = a ( x − h ) 2 + k y = a(x - h)^2 + k y = a ( x − h ) 2 + k Vertex coordinates: h = − b 2 a h = -\frac{b}{2a} h = − 2 a b , k = c − b 2 4 a = f ( h ) k = c - \frac{b^2}{4a} = f(h) k = c − 4 a b 2 = f ( h ) . If a > 0 a > 0 a > 0 , vertex is a minimum at y = k y = k y = k . If a < 0 a < 0 a < 0 , vertex is a maximum at y = k y = k y = k . 2. Polynomial Theorems & Root Analysis For a polynomial P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 : Remainder Theorem: Dividing P ( x ) P(x) P ( x ) by ( x − c ) (x - c) ( x − c ) leaves remainder R = P ( c ) R = P(c) R = P ( c ) . Factor Theorem: ( x − c ) (x - c) ( x − c ) is a factor of P ( x ) P(x) P ( x ) if and only if P ( c ) = 0 P(c) = 0 P ( c ) = 0 . Rational Root Theorem: Any rational root p q \frac{p}{q} q p (in lowest terms) must have p p p as a factor of constant term a 0 a_0 a 0 and q q q as a factor of leading coefficient a n a_n a n . Descartes' Rule of Signs: The number of positive real roots equals the number of sign variations in P ( x ) P(x) P ( x ) or is less by an even integer. 3. Exponential Equations & Applications An exponential function has the form f ( x ) = a ⋅ b x ( b > 0 , b ≠ 1 ) f(x) = a \cdot b^x \quad (b > 0, b \neq 1) f ( x ) = a ⋅ b x ( b > 0 , b  = 1 ) . Compound Interest & Continuous Growth Models Periodic Compounding: A = P ( 1 + r n ) n t A = P \left(1 + \frac{r}{n}\right)^{nt} A = P ( 1 + n r ) n t Continuous Compounding: A = P e r t A = P e^{rt} A = P e r t Radioactive Decay / Half-Life: N ( t ) = N 0 ( 1 2 ) t / t 1 / 2 N(t) = N_0 \left(\frac{1}{2}\right)^{t / t_{1/2}} N ( t ) = N 0 ( 2 1 ) t / t 1/2 4. Logarithmic Properties & Equations Definition: y = log ⁡ b ( x ) ⟺ b y = x ( x > 0 , b > 0 , b ≠ 1 ) y = \log_b(x) \iff b^y = x \quad (x > 0, b > 0, b \neq 1) y = lo g b ( x ) ⟺ b y = x ( x > 0 , b > 0 , b  = 1 ) . Core Logarithmic Laws Product Rule: log ⁡ b ( x y ) = log ⁡ b ( x ) + log ⁡ b ( y ) \log_b(x y) = \log_b(x) + \log_b(y) lo g b ( x y ) = lo g b ( x ) + lo g b ( y ) Quotient Rule: log ⁡ b ( x y ) = log ⁡ b ( x ) − log ⁡ b ( y ) \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) lo g b ( y x ) = lo g b ( x ) − lo g b ( y ) Power Rule: log ⁡ b ( x k ) = k log ⁡ b ( x ) \log_b(x^k) = k \log_b(x) lo g b ( x k ) = k lo g b ( x ) Change of Base Formula: log ⁡ b ( a ) = log ⁡ c ( a ) log ⁡ c ( b ) = ln ⁡ a ln ⁡ b \log_b(a) = \frac{\log_c(a)}{\log_c(b)} = \frac{\ln a}{\ln b} lo g b ( a ) = l o g c ( b ) l o g c ( a ) = l n b l n a Inverse Identities: b log ⁡ b ( x ) = x b^{\log_b(x)} = x b l o g b ( x ) = x and log ⁡ b ( b x ) = x \log_b(b^x) = x lo g b ( b x ) = x CRITICAL ACET TRAP: log ⁡ b ( x + y ) ≠ log ⁡ b ( x ) + log ⁡ b ( y ) \log_b(x + y) \neq \log_b(x) + \log_b(y) lo g b ( x + y )  = lo g b ( x ) + lo g b ( y ) . Logarithms do NOT distribute across addition. Worked Step-by-Step ACET Exam Problems Problem 1: Root Manipulations Using Vieta's Formulas Without explicitly solving the equation 2 x 2 − 8 x + 5 = 0 2x^2 - 8x + 5 = 0 2 x 2 − 8 x + 5 = 0 , find the exact numerical value of r 1 2 + r 2 2 r_1^2 + r_2^2 r 1 2 + r 2 2 , where r 1 r_1 r 1 and r 2 r_2 r 2 are the roots of the equation. Step 1: Identify coefficients. a = 2 a = 2 a = 2 , b = − 8 b = -8 b = − 8 , c = 5 c = 5 c = 5 . Step 2: Apply Vieta's formulas for sum and product. r 1 + r 2 = − − 8 2 = 4 r_1 + r_2 = -\frac{-8}{2} = 4 r 1 + r 2 = − 2 − 8 = 4 r 1 r 2 = 5 2 = 2.5 r_1 r_2 = \frac{5}{2} = 2.5 r 1 r 2 = 2 5 = 2.5 Step 3: Express r 1 2 + r 2 2 r_1^2 + r_2^2 r 1 2 + r 2 2 using identity. r 1 2 + r 2 2 = ( r 1 + r 2 ) 2 − 2 r 1 r 2 r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2 r_1 r_2 r 1 2 + r 2 2 = ( r 1 + r 2 ) 2 − 2 r 1 r 2 r 1 2 + r 2 2 = ( 4 ) 2 − 2 ( 2.5 ) = 16 − 5 = 11 r_1^2 + r_2^2 = (4)^2 - 2(2.5) = 16 - 5 = 11 r 1 2 + r 2 2 = ( 4 ) 2 − 2 ( 2.5 ) = 16 − 5 = 11 Problem 2: Solving an Exponential Equation via Common Bases Solve for x x x : 9 x + 1 − 28 ⋅ 3 x + 3 = 0 9^{x + 1} - 28 \cdot 3^x + 3 = 0 9 x + 1 − 28 ⋅ 3 x + 3 = 0 . Step 1: Express terms with a common base of 3. Notice 9 x + 1 = ( 3 2 ) x + 1 = 3 2 x + 2 = 3 2 ⋅ ( 3 x ) 2 = 9 ( 3 x ) 2 9^{x + 1} = (3^2)^{x + 1} = 3^{2x + 2} = 3^2 \cdot (3^x)^2 = 9 (3^x)^2 9 x + 1 = ( 3 2 ) x + 1 = 3 2 x + 2 = 3 2 ⋅ ( 3 x ) 2 = 9 ( 3 x ) 2 . 9 ( 3 x ) 2 − 28 ( 3 x ) + 3 = 0 9 (3^x)^2 - 28 (3^x) + 3 = 0 9 ( 3 x ) 2 − 28 ( 3 x ) + 3 = 0 Step 2: Substitute u = 3 x u = 3^x u = 3 x (where u > 0 u > 0 u > 0 ). 9 u 2 − 28 u + 3 = 0 9 u^2 - 28 u + 3 = 0 9 u 2 − 28 u + 3 = 0 Step 3: Factor the quadratic in u u u . ( 9 u − 1 ) ( u − 3 ) = 0 ⟹ u = 1 9 or u = 3 (9u - 1)(u - 3) = 0 \implies u = \frac{1}{9} \quad \text{or} \quad u = 3 ( 9 u − 1 ) ( u − 3 ) = 0 ⟹ u = 9 1 or u = 3 Step 4: Solve back for x x x . If 3 x = 1 9 = 3 − 2 ⟹ x = − 2 3^x = \frac{1}{9} = 3^{-2} \implies x = -2 3 x = 9 1 = 3 − 2 ⟹ x = − 2 . If 3 x = 3 1 ⟹ x = 1 3^x = 3^1 \implies x = 1 3 x = 3 1 ⟹ x = 1 . Both solutions x = − 2 x = -2 x = − 2 and x = 1 x = 1 x = 1 are valid. ACET Speed Tactics & Traps Summary

Problem Type Common Student Trap ACET Speed Tactic Quadratic Roots Calculating discriminant when only sum of roots is needed Use Vieta's formula: r 1 + r 2 = − b / a r_1 + r_2 = -b/a r 1 + r 2 = − b / a Log Distribution Writing log ⁡ ( a + b ) = log ⁡ a + log ⁡ b \log(a+b) = \log a + \log b lo g ( a + b ) = lo g a + lo g b Log laws apply to products, not sums Remainder Theorem Performing full long division Evaluate P ( c ) P(c) P ( c ) directly for linear divisor ( x − c ) (x-c) ( x − c ) Exponential Base Forgetting to check u > 0 u > 0 u > 0 when substituting u = a x u = a^x u = a x Discard negative u u u values immediately

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If log_2(x) + log_2(x - 6) = 4, what is the value of x?

-2

8 Check Answer Need a hint? AI

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What is the remainder when the polynomial P(x) = 2x^4 - 3x^3 + 5x - 8 is divided by (x + 2)?

-18

-38 Check Answer Need a hint? AI

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What is the maximum vertical height achieved by a projectile whose trajectory is modeled by h(t) = -5t^2 + 30t + 12 meters?

47 meters

52 meters

57 meters

62 meters Check Answer Need a hint? AI

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