Practice Test 8 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

Convert $4\left(\cos 270^\circ+i\sin 270^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$ .
The quadratic $f(x) = ax^2 + bx + c$ has roots $r_1 = 2 + 3i$ and $r_2 = 2 - 3i$ . If $f(0) = 39$ , what is the value of the leading coefficient $a$ ?
The expression $5(x - 2)^2 - 3(x + 1)(x - 2)$ contains the factor $(x - 2)$ . What is the other factor when this expression is completely factored?
A community pool has two pumps that can fill it. Pump A can fill the pool in 6 hours, and Pump B can fill the pool in 8 hours. Write and solve a rational equation to find how long it takes to fill the pool if both pumps run at the same time. Let $t$ = the time (in hours) to fill the pool together.
Find all points where the line intersects the circle (solve the system). $\begin{cases} y = x \\ x^2 + y^2 = 8 \end{cases}$
Two savings plans are described below. Plan 1: Start with $2000 and add $50 each month. Plan 2: Start with $2000 and increase the balance by 2% each month. Which plan shows constant percent change (exponential), and which shows constant additive change (linear)?
Determine whether the system is consistent. If it has a solution, give it; otherwise state no solution: $\begin{cases} 3x - y = 2\\ 6x - 2y = 5 \end{cases}$
A medicine dose remaining in the bloodstream is measured every 6 hours: Time (hours): 0, 6, 12, 18 Amount (mg): 80, 60, 45, 33.75 Is this constant percent decay? If so, what is the percent decrease per 6 hours?
A medication amount in the bloodstream is modeled by $M(t)=80(0.92)^t$ , where $t$ is in hours. What percent does the amount change each hour, and is it growth or decay?
Consider the functions $p(x) = \frac{3x - 2}{x + 1}$ and $q(x) = \frac{-x - 2}{x - 3}$ . When computing $p(q(x))$ to verify if these are inverse functions, which expression represents the correct first step in the composition?
A bacteria culture doubles every hour. One model is $N(t)=N_0\,2^{t}$ where $t$ is in hours. Rewrite $2^{t}$ to show an equivalent form with a per-minute factor raised to the number of minutes. (There are 60 minutes in an hour.)
Two structural views are possible. Use structure to factor completely: $x^4-1$
Verify that the exponential function $g(x)=2\cdot 3^x$ has constant ratios over unit intervals. Which statement correctly shows the constant ratio for $\Delta x=1$ ?
A force is modeled as the complex number $F=3+4i$ , where $3$ is the horizontal component and $4$ is the vertical component. What is the magnitude of the force (the modulus $|F|$ )?
Let $z_1=-6+4i$ and $z_2=2-2i$ be points on the complex plane. Using the midpoint formula $\frac{z_1+z_2}{2}=\left(\frac{a+c}{2}\right)+\left(\frac{b+d}{2}\right)i,$ what is the midpoint of the segment joining $z_1$ and $z_2$ ?
Synthetic division is an efficient method for dividing by $(x-c)$ : write $c$ on the left, list coefficients on the right, then repeatedly bring down, multiply by $c$ , and add. Use synthetic division to divide $P(x)=x^3+2x^2-5x+3$ by $(x-2).$ (Use $c=2$ with coefficients $1,2,-5,3$ .) Read the quotient from the bottom row and the last value as the remainder (which equals $P(2)$ ). Which result is correct?
Solve for $x$ : $\log_{10}(x)=2$ .
In geometry, the volume of a cone is $V=\dfrac{1}{3}\pi r^2h$ , where $r$ is radius and $h$ is height. Solve $V=\dfrac{1}{3}\pi r^2h$ for $r$ .
Subtract and write the result in standard form (be careful to distribute the negative sign): $\,(3x^4-x^2+6x-9)-(x^4+2x^3-5x+4)\,$ .
Derive the quadratic formula by completing the square on the general quadratic equation $ax^2 + bx + c = 0 \quad (a \ne 0).$ Which expression for $x$ results?

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