Question

solve the equation below. give your answer(s) in simplest form.
$12y^2 - 17y - 5 = 0$
$y = $

1. solve the equation below. give your answer(s) in simplest form.

$r^2 + 15r + 50 = 6 - r$
$r = $

1. solve the equation below. give your answer(s) in simplest form.

$\frac{1}{3}v^2 + 27 = 12$
$v = $

1. solve the equation below. give your answer(s) in simplest form.

$5n^2 = 12n - 8$
$n = $

1. standing from the top of a platform, frank shot an arrow vertically into the air at an initial velocity of 118 ft/s. the height of the arrow, $h$, at $t$ seconds is modeled by the equation $h = -16t^2 + 118t + 15$. how many seconds will it take the arrow to reach the ground?

$t = $

1. solve the equation below. give your answer(s) in simplest form.

$17 = 11 + 2sqrt{16 - 5p}$
$p = $

Explanation:

Step1: Solve $12y^2 -17y -5=0$ via quadratic formula

Quadratic formula: $y=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=12, b=-17, c=-5$
$\sqrt{(-17)^2-4(12)(-5)}=\sqrt{289+240}=\sqrt{529}=23$
$y=\frac{17\pm23}{24}$

Step2: Calculate two $y$ values

$y_1=\frac{17+23}{24}=\frac{40}{24}=\frac{5}{3}$, $y_2=\frac{17-23}{24}=\frac{-6}{24}=-\frac{1}{4}$

Step3: Rearrange $r^2+15r+50=6-r$ to standard form

$r^2+16r+44=0$

Step4: Solve $r^2+16r+44=0$ via quadratic formula

$a=1, b=16, c=44$
$\sqrt{16^2-4(1)(44)}=\sqrt{256-176}=\sqrt{80}=4\sqrt{5}$
$r=\frac{-16\pm4\sqrt{5}}{2}=-8\pm2\sqrt{5}$

Step5: Solve $\frac{1}{3}v^2+27=12$ for $v$

$\frac{1}{3}v^2=12-27=-15$
$v^2=-45$
$v=\pm3i\sqrt{5}$

Step6: Rearrange $5n^2=12n-8$ to standard form

$5n^2-12n+8=0$

Step7: Solve $5n^2-12n+8=0$ via quadratic formula

$a=5, b=-12, c=8$
$\sqrt{(-12)^2-4(5)(8)}=\sqrt{144-160}=\sqrt{-16}=4i$
$n=\frac{12\pm4i}{10}=\frac{6\pm2i}{5}$

Step8: Solve $h=-16t^2+118t+15=0$

$a=-16, b=118, c=15$
$\sqrt{118^2-4(-16)(15)}=\sqrt{13924+960}=\sqrt{14884}=122$
$t=\frac{-118\pm122}{-32}$
Take positive root: $t=\frac{-118+122}{-32}$ is negative, discard; $t=\frac{-118-122}{-32}=\frac{-240}{-32}=7.5$

Step9: Solve $17=11+2\sqrt[4]{16-5p}$

Isolate radical: $2\sqrt[4]{16-5p}=17-11=6$
$\sqrt[4]{16-5p}=3$
Raise to 4th power: $16-5p=81$
$-5p=81-16=65$
$p=-13$

Answer:

$y = \frac{5}{3}, -\frac{1}{4}$
$r = -8+2\sqrt{5}, -8-2\sqrt{5}$
$v = 3i\sqrt{5}, -3i\sqrt{5}$
$n = \frac{6+2i}{5}, \frac{6-2i}{5}$
$t = 7.5$
$p = -13$