🦕 Dino Algebra 2

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Unit 1

Quadratic Functions & Equations

Standard Form\(f(x)=ax^2+bx+c\) — vertex at \(x=-\frac{b}{2a}\)

Vertex Form\(f(x)=a(x-h)^2+k\) — vertex \((h,k)\)

Discriminant\(\Delta=b^2-4ac\): 2 real, 1 real, or 2 complex roots

Completing SquareAdd/subtract \(\left(\frac{b}{2}\right)^2\) to both sides

🔑 Memorize — Quadratic Formula \[x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\]

Example

Solve \(2x^2-5x-3=0\).
\(a=2,b=-5,c=-3\) → \(\Delta=25+24=49\)

✅ \(x=\dfrac{5\pm7}{4}\) → \(x=3\) or \(x=-\tfrac{1}{2}\)

Unit 2

Complex Numbers

Definition\(i=\sqrt{-1}\), \(i^2=-1\), \(i^3=-i\), \(i^4=1\)

OperationsAdd real + real, imaginary + imaginary. Multiply using FOIL.

ConjugateConjugate of \(a+bi\) is \(a-bi\). Product: \(a^2+b^2\)

Modulus\(|a+bi|=\sqrt{a^2+b^2}\)

🔑 Memorize \(\dfrac{a+bi}{c+di}=\dfrac{(a+bi)(c-di)}{c^2+d^2}\)

Example

Simplify \(\dfrac{3+2i}{1-i}\).

✅ Multiply by \(\dfrac{1+i}{1+i}\): \(\dfrac{(3+2i)(1+i)}{2}=\dfrac{1+5i}{2}=\tfrac{1}{2}+\tfrac{5}{2}i\)

Unit 3

Polynomial Functions

Factor Theorem\((x-c)\) is a factor ↔ \(f(c)=0\)

Remainder TheoremDivide \(f(x)\) by \((x-c)\) → remainder is \(f(c)\)

Rational Root ThmPossible roots: \(\pm\frac{p}{q}\) where \(p|\text{const},\,q|\text{lead}\)

End BehaviorDetermined by degree & leading coefficient sign

🔑 Memorize — Sum & Difference of Cubes \[a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)\]

Example

Find all roots of \(f(x)=x^3-6x^2+11x-6\).

✅ Rational roots: ±1,±2,±3,±6. Test \(x=1\): \(1-6+11-6=0\) ✓. Factor: \((x-1)(x-2)(x-3)\). Roots: \(x=1,2,3\).

Unit 4

Rational Functions

Vertical AsymptoteSet denominator \(=0\) (after canceling common factors)

Horizontal AsymptoteCompare degrees of numerator vs. denominator

HoleOccurs where a factor cancels in both num. and denom.

HA Rulesdeg(N)<deg(D): \(y=0\); equal: \(y=\frac{\text{lead}_N}{\text{lead}_D}\); greater: none (oblique)

Example

\(f(x)=\dfrac{x^2-4}{x^2-5x+6}\): find asymptotes and holes.

✅ \(\frac{(x-2)(x+2)}{(x-2)(x-3)}\). Hole at \(x=2\); VA: \(x=3\); HA: \(y=1\).

Unit 5

Radical & Exponential Functions

Exponent Rules\(a^m\cdot a^n=a^{m+n}\), \(\frac{a^m}{a^n}=a^{m-n}\), \((a^m)^n=a^{mn}\)

Rational Exponent\(a^{m/n}=\left(\sqrt[n]{a}\right)^m\)

Exponential Growth\(A=P(1+r)^t\); Decay: \(A=P(1-r)^t\)

Natural Base\(e\approx2.718\); \(A=Pe^{rt}\)

🔑 Memorize \[a^{-n}=\dfrac{1}{a^n}\qquad \sqrt[n]{a^m}=a^{m/n}\]

Example

Simplify \(\dfrac{x^{1/2}\cdot x^{2/3}}{x^{1/6}}\).

✅ Exponent: \(\frac{1}{2}+\frac{2}{3}-\frac{1}{6}=\frac{3+4-1}{6}=1\). Answer: \(x^1=x\).

Unit 6

Logarithms

Definition\(\log_b a=c \iff b^c=a\)

Product Rule\(\log_b(mn)=\log_b m+\log_b n\)

Quotient Rule\(\log_b\!\frac{m}{n}=\log_b m-\log_b n\)

Power Rule\(\log_b(m^p)=p\log_b m\)

🔑 Memorize — Change of Base \[\log_b a=\dfrac{\ln a}{\ln b}=\dfrac{\log a}{\log b}\]

Example

Solve \(\log_3(x+1)+\log_3(x-1)=2\).

✅ \(\log_3[(x+1)(x-1)]=2\) → \(x^2-1=9\) → \(x^2=10\) → \(x=\sqrt{10}\) (reject negative since \(x>1\)).

Unit 7

Sequences & Series

Arithmetic\(a_n=a_1+(n-1)d\); Sum: \(S_n=\frac{n}{2}(a_1+a_n)\)

Geometric\(a_n=a_1\cdot r^{n-1}\); Sum: \(S_n=\frac{a_1(1-r^n)}{1-r}\)

Infinite Geo Sum\(S=\frac{a_1}{1-r}\) when \(|r|<1\)

Sigma Notation\(\displaystyle\sum_{k=1}^{n}k=\frac{n(n+1)}{2}\)

Example

Find \(S_8\) of the geometric series \(3, 6, 12, \ldots\)

✅ \(a_1=3, r=2\). \(S_8=\frac{3(1-2^8)}{1-2}=\frac{3(-255)}{-1}=765\).

Unit 8

Matrices & Systems

Matrix Mult.Row × Column. \([A]_{m\times n}\cdot[B]_{n\times p}=[C]_{m\times p}\)

Determinant 2×2\(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\)

Inverse 2×2\(A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)

Cramer's Rule\(x=\frac{D_x}{D},\;y=\frac{D_y}{D}\)

Example

Find the inverse of \(\begin{pmatrix}3&1\\5&2\end{pmatrix}\).

✅ \(\det=3(2)-1(5)=1\). \(A^{-1}=\begin{pmatrix}2&-1\\-5&3\end{pmatrix}\).

Unit 9

Conic Sections

Circle\((x-h)^2+(y-k)^2=r^2\), center \((h,k)\)

Parabola (vert.)\(y=a(x-h)^2+k\); focus at distance \(\frac{1}{4a}\)

Ellipse\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\); \(a>b\), foci on major axis

Hyperbola\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\); asymptotes \(y=\pm\frac{b}{a}x\)

Example

Write in standard form: \(x^2+y^2-6x+4y+4=0\).

✅ Complete square: \((x-3)^2+(y+2)^2=9\). Circle, center \((3,-2)\), radius \(3\).

Unit 10

Probability & Statistics

Permutation\(P(n,r)=\dfrac{n!}{(n-r)!}\)

Combination\(C(n,r)=\dfrac{n!}{r!(n-r)!}\)

Binomial Theorem\((a+b)^n=\displaystyle\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)

Std. Deviation\(\sigma=\sqrt{\frac{\sum(x_i-\mu)^2}{n}}\)

Example

Expand the 3rd term of \((2x+3)^5\).

✅ 3rd term (\(k=2\)): \(\binom{5}{2}(2x)^3(3)^2=10\cdot8x^3\cdot9=720x^3\).

Rex says: Great job reviewing! 🦕 Now switch to the Problems tab and show me what you've got! Remember — read each question carefully before choosing. You've totally got this! 💪

Q1 Quadratic · Medium

The vertex of \(f(x)=3x^2-12x+7\) is at the point:

📖 Solution

Vertex x-coordinate: \(x=-\dfrac{b}{2a}=-\dfrac{-12}{2(3)}=\dfrac{12}{6}=2\).
Substitute: \(f(2)=3(4)-12(2)+7=12-24+7=-5\).
∴ Vertex is \((2,\ -5)\).

Q2 Quadratic · Hard

The equation \(x^2-4x+k=0\) has two complex (non-real) roots. Which of the following must be true?

📖 Solution

For complex (non-real) roots the discriminant must be negative.
\(\Delta=b^2-4ac=(-4)^2-4(1)(k)=16-4k\).
We need \(16-4k<0\) → \(4k>16\) → \(k>4\).

Q3 Complex Numbers · Medium

What is \((3+4i)(3-4i)\)?

📖 Solution

\((3+4i)(3-4i)=3^2-(4i)^2=9-16i^2=9-16(-1)=9+16=25\).
Note: (C) is a partially simplified expression, not a simplified answer. The correct simplified value is \(25\).

Q4 Complex Numbers · Hard

Simplify \(i^{47}\).

📖 Solution

Powers of \(i\) cycle with period 4: \(i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1\).
\(47\div4=11\) remainder \(3\).
Therefore \(i^{47}=i^3=-i\).

Q5 Polynomials · Medium

When \(f(x)=2x^3-5x^2+3x-7\) is divided by \((x-2)\), the remainder is:

📖 Solution

By the Remainder Theorem, the remainder = \(f(2)\).
\(f(2)=2(8)-5(4)+3(2)-7=16-20+6-7=-5\).

Q6 Polynomials · Hard

How many positive real zeros does \(f(x)=x^4-3x^3+x^2+3x-2\) have, according to Descartes' Rule of Signs?

📖 Solution

Count sign changes in the coefficients: \(+1,\,-3,\,+1,\,+3,\,-2\).
Sign changes: \(+→-\) (1), \(-→+\) (2), \(+→+\) (no), \(+→-\) (3).
There are 3 sign changes, so there are 3 or 1 positive real zeros.

Q7 Rational Functions · Medium

Which of the following is the horizontal asymptote of \(f(x)=\dfrac{4x^2-1}{2x^2+3x}\)?

📖 Solution

Degrees of numerator and denominator are both 2 (equal).
Horizontal asymptote = ratio of leading coefficients: \(\dfrac{4}{2}=2\).
∴ \(y=2\).

Q8 Radical & Exponential · Medium

Solve for \(x\): \(4^{x+1}=8^{x-1}\).

📖 Solution

Write both sides as powers of 2: \(4=2^2\) and \(8=2^3\).
\(2^{2(x+1)}=2^{3(x-1)}\) → \(2x+2=3x-3\) → \(x=5\).

Q9 Logarithms · Medium

Evaluate: \(\log_4 128\).

📖 Solution

Express in base 2: \(4=2^2,\;128=2^7\).
\(\log_4 128=\dfrac{\log_2 128}{\log_2 4}=\dfrac{7}{2}\).

Q10 Logarithms · Hard

Solve: \(\log_2(x)+\log_2(x-6)=4\).

📖 Solution

\(\log_2[x(x-6)]=4\) → \(x(x-6)=16\) → \(x^2-6x-16=0\).
Factor: \((x-8)(x+2)=0\) → \(x=8\) or \(x=-2\).
Domain: \(x>0\) and \(x>6\), so \(x>6\). Reject \(x=-2\).
∴ \(x=8\).

Q11 Sequences · Medium

The sum of the first 10 terms of an arithmetic sequence with \(a_1=3\) and common difference \(d=4\) is:

📖 Solution

\(a_{10}=3+(10-1)(4)=3+36=39\).
\(S_{10}=\dfrac{10}{2}(a_1+a_{10})=5(3+39)=5(42)=210\).

Q12 Series · Hard

What is the sum of the infinite geometric series \(8-4+2-1+\cdots\)?

📖 Solution

\(a_1=8,\;r=-\dfrac{1}{2}\). Since \(|r|=\dfrac{1}{2}<1\), the series converges.
\(S=\dfrac{a_1}{1-r}=\dfrac{8}{1-(-\frac{1}{2})}=\dfrac{8}{\frac{3}{2}}=\dfrac{16}{3}\).

Q13 Matrices · Medium

If \(A=\begin{pmatrix}2&-1\\4&3\end{pmatrix}\), what is \(\det(A)\)?

📖 Solution

\(\det(A)=ad-bc=(2)(3)-(-1)(4)=6+4=10\).

Q14 Matrices · Hard

Using Cramer's Rule, solve for \(y\) in the system: \(2x+y=5\) and \(x-3y=-1\).

📖 Solution

\(D=\begin{vmatrix}2&1\\1&-3\end{vmatrix}=(2)(-3)-(1)(1)=-7\).
\(D_y=\begin{vmatrix}2&5\\1&-1\end{vmatrix}=(2)(-1)-(5)(1)=-7\).
\(y=\dfrac{D_y}{D}=\dfrac{-7}{-7}=1\).

Q15 Conics · Medium

The equation \(\dfrac{(x-2)^2}{16}+\dfrac{(y+1)^2}{9}=1\) represents an ellipse. What is the length of its major axis?

📖 Solution

From the standard form \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\):
Here \(a^2=16\), so \(a=4\). The major axis has length \(2a=2(4)=8\).

Q16 Conics · Hard

The hyperbola \(\dfrac{x^2}{9}-\dfrac{y^2}{16}=1\) has asymptotes:

📖 Solution

For hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\), asymptotes are \(y=\pm\dfrac{b}{a}x\).
Here \(a^2=9\Rightarrow a=3\) and \(b^2=16\Rightarrow b=4\).
∴ Asymptotes: \(y=\pm\dfrac{4}{3}x\).

Q17 Probability · Medium

How many different 4-letter arrangements (order matters) can be made from the letters in "MATH"?

📖 Solution

All 4 letters in "MATH" are distinct, and we arrange all 4.
\(P(4,4)=4!=4\times3\times2\times1=24\).

Q18 Binomial Theorem · Hard

What is the coefficient of \(x^3\) in the expansion of \((x-2)^5\)?

📖 Solution

General term: \(\binom{5}{k}(x)^{5-k}(-2)^k\). For \(x^3\), we need \(5-k=3\), so \(k=2\).
Term: \(\binom{5}{2}(x)^3(-2)^2=10\cdot x^3\cdot4=40x^3\).
∴ Coefficient of \(x^3\) is \(40\).

Q19 Exponential Functions · Hard

A population doubles every 5 years. Starting with 1,000 people, how many will there be after 20 years?

📖 Solution

The number of doubling periods in 20 years: \(\dfrac{20}{5}=4\).
Population \(=1000\times2^4=1000\times16=16{,}000\).

Q20 Rational Functions · Hard

Which value of \(x\) is NOT in the domain of \(f(x)=\dfrac{x+3}{x^2-x-6}\)?

📖 Solution

Factor the denominator: \(x^2-x-6=(x-3)(x+2)\).
The denominator equals zero when \(x=3\) or \(x=-2\).
Note: \(x=-3\) is NOT excluded (the numerator \(x+3\) becomes 0 there, creating a hole, but the denominator ≠ 0 at \(x=-3\)).
Wait — let's recheck: at \(x=-3\): \((-3)^2-(-3)-6=9+3-6=6\neq0\). So \(x=-3\) IS in the domain.
∴ Values NOT in the domain: \(x=3\) and \(x=-2\).

📖 Answer Key & Explanations

Q1 — Quadratic Vertex

✅ Correct Answer: A) (2, −5)

Vertex x-coord: \(x=-\frac{-12}{6}=2\). Then \(f(2)=12-24+7=-5\). Vertex: \((2,-5)\).

Q2 — Discriminant

✅ Correct Answer: C) k > 4

\(\Delta=16-4k<0 \Rightarrow k>4\) for two complex roots.

Q3 — Complex Conjugate Product

✅ Correct Answer: A) 25

\((3+4i)(3-4i)=9-16i^2=9+16=25\).

Q4 — Powers of i

✅ Correct Answer: D) −i

\(47=4(11)+3\), so \(i^{47}=i^3=-i\).

Q5 — Remainder Theorem

✅ Correct Answer: B) −5

\(f(2)=16-20+6-7=-5\).

Q6 — Descartes' Rule

✅ Correct Answer: B) 3 or 1

Coefficients \(+,-,+,+,-\) give 3 sign changes → 3 or 1 positive real zeros.

Q7 — Horizontal Asymptote

✅ Correct Answer: B) y = 2

Equal degrees → HA = leading coefficient ratio = \(4/2=2\).

Q8 — Exponential Equation

✅ Correct Answer: B) x = 5

\(2^{2(x+1)}=2^{3(x-1)} \Rightarrow 2x+2=3x-3 \Rightarrow x=5\).

Q9 — Logarithm Evaluation

✅ Correct Answer: C) 7/2

\(\log_4 128=\frac{\log_2 128}{\log_2 4}=\frac{7}{2}\).

Q10 — Logarithm Equation

✅ Correct Answer: A) x = 8

\(x(x-6)=16 \Rightarrow x^2-6x-16=0 \Rightarrow x=8\) or \(-2\). Reject \(x=-2\) (domain).

Q11 — Arithmetic Series Sum

✅ Correct Answer: C) 210

\(a_{10}=39\). \(S_{10}=5(3+39)=210\).

Q12 — Infinite Geometric Series

✅ Correct Answer: A) 16/3

\(r=-\frac{1}{2}\), \(|r|<1\). \(S=\frac{8}{1+\frac{1}{2}}=\frac{8}{\frac{3}{2}}=\frac{16}{3}\).

Q13 — Matrix Determinant

✅ Correct Answer: B) 10

\(\det=2(3)-(-1)(4)=6+4=10\).

Q14 — Cramer's Rule

✅ Correct Answer: A) y = 1

\(D=-7,\;D_y=-7\Rightarrow y=1\).

Q15 — Ellipse Major Axis

✅ Correct Answer: D) 8

\(a^2=16\Rightarrow a=4\). Major axis length \(=2a=8\).

Q16 — Hyperbola Asymptotes

✅ Correct Answer: B) y = ±(4/3)x

For \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\): asymptotes \(y=\pm\frac{b}{a}x=\pm\frac{4}{3}x\).

Q17 — Permutations

✅ Correct Answer: C) 24

\(P(4,4)=4!=24\).

Q18 — Binomial Theorem

✅ Correct Answer: B) 40

\(k=2\): \(\binom{5}{2}(-2)^2\cdot x^3=10\cdot4\cdot x^3=40x^3\). Coefficient is \(40\).

Q19 — Exponential Growth

✅ Correct Answer: C) 16,000

\(1000\times2^{20/5}=1000\times2^4=16000\).

Q20 — Rational Function Domain

✅ Correct Answer: C) x = 3 and x = −2

Denominator \(=(x-3)(x+2)=0\) at \(x=3\) and \(x=-2\). Both excluded from domain.