Q: A biologist is studying the growth of a bacteria population that doubles every 3 hours. If the initial population is 500, which equation models the population $$ P $$ after $$ t $$ hours?

$$ P = 500 \times 2$$^{t/3}$$ $$

Question 1

Which of the following expressions is equivalent to $$ (2x^3$y^2)^2$ $$?

Accepted Answer

$$ 4x^6$y^4$$ $

Question 2

A biologist is studying the growth of a bacteria population that doubles every 3 hours. If the initial population is 500, which equation models the population $$ P $$ after $$ t $$ hours?

Accepted Answer

$$ P = 500 	imes 2$$^{t/3}$$ $$

Question 3

Simplify the expression $$ \dfrac{3x^2$y^{-3}$$}{$$6x^{-1}y^2$} .$$

Accepted Answer

$$ \dfrac{1}{2}x^3$y^{-5}$$ $

Question 4

Which of the following is the correct factorization of $$ x^2$ - 9 $$?

Accepted Answer

$$ (x + 3)(x - 3) $$

Question 5

A company’s profit $$ P  (in $$thousands of dollars) is modeled by the quadratic equation $$ P = -2x^2$ + 8x + 10 $$, where $$ x  is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

$$ 26 $$ thousand dollars at $$ x = 2 $$ thousand units

Question 6

Which of the following is the solution set for $$ 2x^2$ - 8x = 0 $$?

Accepted Answer

$$ \{0, 4\} $$

Question 7

Simplify the following expression: $$ (3x^2$y^3)^2$ \cdot (2xy)^3$ .$$

Accepted Answer

$$ 72x^7$y^9$$ $

Question 8

A quadratic equation $$ ax^2$ + bx + c = 0 $$ has real and equal roots. Which of the following must be true?

Accepted Answer

$$ b^2$ - 4ac = 0 $$

Question 9

Which of the following is the correct simplification of $$ \sqrt{50} + 2\sqrt{8} $$?

Accepted Answer

$$ 5\sqrt{2} + 4\sqrt{2} = 9\sqrt{2} $$

Question 10

A ball is thrown upward from the ground with an initial velocity of $$ 32  ft/s. $$Its height $$ h  (in $$feet) after $$ t $$ seconds is given by $$ h = -16t^2$ + 32t . At $$what time does the ball reach its maximum height?

Accepted Answer

$$ t = 1 $$ second

Question 11

Which of the following is the correct result of dividing $$ (x^3$ - 8)  by$$ $$ (x - 2) $$?

Accepted Answer

$$ x^2$ + 2x + 4 $$

Question 12

If $$ f(x) = 2x^2$ - 3x + 5 $$, what is $$ f(-2) $$?

Accepted Answer

$$ 19 $$

Question 13

Which of the following is the correct solution to $$ \sqrt{x + 5} = 3 $$?

Accepted Answer

$$ x = 4 $$

Question 14

A rectangular garden has a length that is 3 meters more than twice its width. If the area is $$ 99 $$ square meters, what is the width of the garden?

Accepted Answer

$$ 6 $$ meters

Question 15

Which of the following is the correct simplification of $$ (x^2$ - 4x + 4) $$?

Accepted Answer

$$ (x - 2)^2$ $$

Question 16

If $$ x = 3  is a $$solution to the equation $$ x^2$ - kx + 6 = 0 $$, what is the value of $$ k $$?

Accepted Answer

$$ 5 $$

Question 17

Which of the following is the correct result of $$ (2x - 5)(x + 3) $$?

Accepted Answer

$$ 2x^2$ + x - 15 $$

Question 18

A chemist mixes a solution that is $$ 30\% $$ salt with another that is $$ 10\% $$ salt to make $$ 20 $$ liters of a $$ 15\% $$ salt solution. How many liters of the $$ 30\% $$ solution are needed?

Accepted Answer

$$ 5 $$ liters

Question 19

Which of the following is the correct solution to $$ 2$$^{x}$$ = 16 $$?

Accepted Answer

$$ x = 4 $$

Question 20

If $$ x^2$ + 6x + 9 = 0 $$, what are the solutions for $$ x $$?

Accepted Answer

$$ x = -3 $$

Math 980: Algebraic Problem Solving II – Syllabus and Study Guide

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