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Core Concepts & Formulas
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Key Formulas
Quadratic: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Vieta's: \(r+s = -\dfrac{b}{a},\; rs = \dfrac{c}{a}\)
Difference of squares: \(a^2-b^2=(a-b)(a+b)\)
Vieta's: \(r+s = -\dfrac{b}{a},\; rs = \dfrac{c}{a}\)
Difference of squares: \(a^2-b^2=(a-b)(a+b)\)
Must Memorize
- Discriminant \(\Delta = b^2-4ac\): positive → 2 real roots; zero → 1 repeated root; negative → no real roots
- Sum of roots \(= -b/a\), Product \(= c/a\)
- \((a+b)^2 = a^2+2ab+b^2\)
- Always factor before using quadratic formula
Warm-up Example
If \(x^2 - 5x + 6 = 0\), find the sum and product of the roots.
Answer: Sum \(= -(-5)/1 = 5\); Product \(= 6/1 = 6\). (Roots are 2 and 3.)
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Key Formulas
Number of divisors of \(n = p_1^{a_1}\cdots p_k^{a_k}\):
\(\tau(n) = (a_1+1)(a_2+1)\cdots(a_k+1)\)
GCD × LCM \(= a \times b\)
\(a \equiv b \pmod{m}\) means \(m \mid (a-b)\)
\(\tau(n) = (a_1+1)(a_2+1)\cdots(a_k+1)\)
GCD × LCM \(= a \times b\)
\(a \equiv b \pmod{m}\) means \(m \mid (a-b)\)
Must Memorize
- Divisibility: 3 (digit sum), 9 (digit sum), 11 (alternating sum), 7 (no easy rule)
- Perfect squares end in 0,1,4,5,6,9
- Euclidean Algorithm: \(\gcd(a,b)=\gcd(b, a\bmod b)\)
- Sum of first \(n\) odd numbers \(= n^2\)
Warm-up Example
How many positive divisors does \(360 = 2^3 \cdot 3^2 \cdot 5\) have?
Answer: \(\tau(360) = (3+1)(2+1)(1+1) = 4 \cdot 3 \cdot 2 = 24\) divisors.
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Key Formulas
Area of triangle: \(\dfrac{1}{2}bh\) or \(\dfrac{1}{2}ab\sin C\)
Circle: \(A=\pi r^2\), \(C=2\pi r\)
Pythagorean Theorem: \(a^2+b^2=c^2\)
Heron's: \(A = \sqrt{s(s-a)(s-b)(s-c)}\)
Circle: \(A=\pi r^2\), \(C=2\pi r\)
Pythagorean Theorem: \(a^2+b^2=c^2\)
Heron's: \(A = \sqrt{s(s-a)(s-b)(s-c)}\)
Must Memorize
- Special triangles: 3-4-5, 5-12-13, 8-15-17, 7-24-25
- 30-60-90: sides \(1 : \sqrt{3} : 2\)
- 45-45-90: sides \(1 : 1 : \sqrt{2}\)
- Inscribed angle = half central angle
Warm-up Example
A right triangle has legs 5 and 12. Find its hypotenuse and area.
Answer: Hypotenuse \(=\sqrt{25+144}=13\). Area \(= \frac{1}{2}(5)(12) = 30\).
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Key Formulas
Permutations: \(P(n,r)=\dfrac{n!}{(n-r)!}\)
Combinations: \(\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}\)
Stars and bars: \(\dbinom{n+k-1}{k-1}\)
Combinations: \(\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}\)
Stars and bars: \(\dbinom{n+k-1}{k-1}\)
Must Memorize
- Order matters → Permutation; order doesn't → Combination
- Complement counting: Total − Forbidden
- Pascal's: \(\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}\)
- Multiplication principle: independent choices multiply
Warm-up Example
How many ways can 3 students be chosen from a group of 7 for a committee?
Answer: \(\binom{7}{3} = \dfrac{7!}{3!\,4!} = 35\) ways.
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Key Formulas
\(P(A) = \dfrac{\text{favorable}}{\text{total}}\)
\(P(A \cup B) = P(A)+P(B)-P(A\cap B)\)
Conditional: \(P(A|B) = \dfrac{P(A\cap B)}{P(B)}\)
\(P(A \cup B) = P(A)+P(B)-P(A\cap B)\)
Conditional: \(P(A|B) = \dfrac{P(A\cap B)}{P(B)}\)
Must Memorize
- Independent events: \(P(A\cap B)=P(A)\cdot P(B)\)
- Complementary: \(P(\text{not }A) = 1 - P(A)\)
- Geometric probability: ratio of areas/lengths
- At least one = \(1 - P(\text{none})\)
Warm-up Example
A fair coin is flipped 3 times. What is the probability of getting at least one head?
Answer: \(P(\text{at least one H}) = 1 - P(\text{no heads}) = 1 - \frac{1}{8} = \frac{7}{8}\).
Practice Problems
1
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If \(3(x - 2) + 4 = 2x + 6\), what is the value of \(x\)?
2
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The two roots of \(x^2 - 7x + k = 0\) are both positive integers. What is the sum of all possible values of \(k\)?
3
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How many positive integers less than 100 have exactly 3 positive divisors?
4
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In triangle \(ABC\), \(AB = 10\), \(BC = 10\), and \(\angle ABC = 60°\). What is the area of triangle \(ABC\)?
5
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How many 4-digit positive integers have digits that are all different and sum to 10, where the leading digit is nonzero?
6
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Two fair six-sided dice are rolled. What is the probability that the product of the two numbers rolled is a multiple of 4?
7
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The sum of the first \(n\) terms of a geometric series is \(S_n = 3^n - 1\). What is the 5th term of the series?
8
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A chord of length 8 is drawn in a circle of radius 5. What is the distance from the center of the circle to the chord?
9
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What is the remainder when \(7^{2025}\) is divided by \(10\)?
10
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For how many integer values of \(x\) is \(|2x - 5| \leq 9\) true?
11
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Let \(f(x) = x^2 - 4x + 7\). The minimum value of \(f(x)\) occurs at \(x = c\). What is \(f(c)\)?
12
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How many shortest paths are there from the bottom-left corner to the top-right corner of a \(4 \times 3\) grid (4 units wide, 3 units tall), moving only right or up?
13
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The GCD of two positive integers \(a\) and \(b\) is 12 and their LCM is 180. If \(a = 36\), what is \(b\)?
14
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The line \(y = 2x + 3\) is reflected over the \(y\)-axis. What is the \(x\)-intercept of the reflected line?
15
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A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. Given that the first marble drawn is red, what is the probability that the second marble is also red?
16
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If \(x + y = 10\) and \(xy = 24\), what is the value of \(x^2 + y^2\)?
17
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A right circular cone has a base radius of 6 and a slant height of 10. What is the volume of the cone?
18
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For how many prime numbers \(p\) is \(p^2 + 2\) also prime?
19
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Among integers from 1 to 100 (inclusive), how many are divisible by 3 or 7 (or both)?
20
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Alice can paint a fence in 6 hours, and Bob can paint the same fence in 3 hours. If they work together for 1 hour and then Alice leaves, how many additional hours will it take Bob to finish the fence alone?