Compute the complex quotient $\frac{3+4i}{1-2i}$, and then multiply that result by the conjugate of the denominator $1+2i$.

Solve $(2x+3{)}^2=50$ and simplify the answers.

Solve $3x^2+18x+11=0$ by completing the square.

Solve $\frac{1}{2}{x}^2+2x-3=0$ using the quadratic formula and simplify your answers.

Convert the quadratic: y = x² − 6x + 5 into vertex form by completing the square, and identify the vertex.

Which single sentence most precisely defines a one-to-one function (injective)?

Solve $5^{2x-1}=125$ for $x$.

A population grows according to $P\left(t\right)=200\cdot 2^{\frac{t}{3}}$, where $t$ is in months. Solve for $t$ when $P\left(t\right)=1600$.

Solve for $x$: ${\log }_2\left(x^2\right)-{\log }_2(x-1)=3$

Convert $x^2+y^2+\frac{1}{3}x-\frac{5}{2}y-1=0$ into standard form.

Design an elliptical garden with major axis length $20$ and minor axis length $12$, centered at the origin with horizontal major axis. Provide the standard-form equation and the distance from the center to each focus.

Convert the equation $y=2x^2+8x+5$ into vertex form by completing the square, and identify the vertex.

Find the equations of the asymptotes of the hyperbola $\frac{{(x-1)}^2}{16}-\frac{{(y+2)}^2}{9}=1$.

A bacterial colony starts with $50$ bacteria and doubles every hour. Let $B_n$ represent the $n$^th term of the sequence, where $B_1$ is the initial count. Write an explicit formula and find $B_6$.

Two arithmetic sequences are given by $a_n=6+4(n-1)$ and $b_n=2+6(n-1)$. Find the smallest positive integer $n$ such that $a_n=b_n$.

A biologist models a population by $P_n=1,000,000\cdot (0.7{)}^n$ where $n$ is hours. Determine after how many whole hours the population will drop below $1,000$ individuals. Determine the minimal integer $n$.

Which of these is an algebraic simplification of $(n+2)!-n!$ for integer $n\ge 0$?