The problem asks to solve six quadratic equations using the quadratic formula. The equations are: 1) $4n^2 = -12n - 2$ 2) $9p^2 - 24 = -5p$ 3) $-3n^2 + 5n = -14$ 4) $4n^2 + 2n = 9$ 5) $-7r^2 = -12$ 6) $-8r^2 = -7r - 5$

1. Problem Description

The problem asks to solve six quadratic equations using the quadratic formula. The equations are:

1) $4n^2 = -12n - 2$

2) $9p^2 - 24 = -5p$

3) $-3n^2 + 5n = -14$

4) $4n^2 + 2n = 9$

5) $-7r^2 = -12$

6) $-8r^2 = -7r - 5$

2. Solution Steps

First, we rewrite each equation in the standard quadratic form $ax^2 + bx + c = 0$. Then we use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

1) $4n^2 = -12n - 2$

$4n^2 + 12n + 2 = 0$

$a = 4, b = 12, c = 2$

$n = \frac{-12 \pm \sqrt{12^2 - 4(4)(2)}}{2(4)}$

$n = \frac{-12 \pm \sqrt{144 - 32}}{8}$

$n = \frac{-12 \pm \sqrt{112}}{8}$

$n = \frac{-12 \pm 4\sqrt{7}}{8}$

$n = \frac{-3 \pm \sqrt{7}}{2}$

2) $9p^2 - 24 = -5p$

$9p^2 + 5p - 24 = 0$

$a = 9, b = 5, c = -24$

$p = \frac{-5 \pm \sqrt{5^2 - 4(9)(-24)}}{2(9)}$

$p = \frac{-5 \pm \sqrt{25 + 864}}{18}$

$p = \frac{-5 \pm \sqrt{889}}{18}$

3) $-3n^2 + 5n = -14$

$-3n^2 + 5n + 14 = 0$

$3n^2 - 5n - 14 = 0$

$a = 3, b = -5, c = -14$

$n = \frac{5 \pm \sqrt{(-5)^2 - 4(3)(-14)}}{2(3)}$

$n = \frac{5 \pm \sqrt{25 + 168}}{6}$

$n = \frac{5 \pm \sqrt{193}}{6}$

4) $4n^2 + 2n = 9$

$4n^2 + 2n - 9 = 0$

$a = 4, b = 2, c = -9$

$n = \frac{-2 \pm \sqrt{2^2 - 4(4)(-9)}}{2(4)}$

$n = \frac{-2 \pm \sqrt{4 + 144}}{8}$

$n = \frac{-2 \pm \sqrt{148}}{8}$

$n = \frac{-2 \pm 2\sqrt{37}}{8}$

$n = \frac{-1 \pm \sqrt{37}}{4}$

5) $-7r^2 = -12$

$-7r^2 + 12 = 0$

$7r^2 - 12 = 0$

$a = 7, b = 0, c = -12$

$r = \frac{0 \pm \sqrt{0^2 - 4(7)(-12)}}{2(7)}$

$r = \frac{\pm \sqrt{336}}{14}$

$r = \frac{\pm 4\sqrt{21}}{14}$

$r = \frac{\pm 2\sqrt{21}}{7}$

6) $-8r^2 = -7r - 5$

$-8r^2 + 7r + 5 = 0$

$8r^2 - 7r - 5 = 0$

$a = 8, b = -7, c = -5$

$r = \frac{7 \pm \sqrt{(-7)^2 - 4(8)(-5)}}{2(8)}$

$r = \frac{7 \pm \sqrt{49 + 160}}{16}$

$r = \frac{7 \pm \sqrt{209}}{16}$

3. Final Answer

1) $n = \frac{-3 \pm \sqrt{7}}{2}$

2) $p = \frac{-5 \pm \sqrt{889}}{18}$

3) $n = \frac{5 \pm \sqrt{193}}{6}$

4) $n = \frac{-1 \pm \sqrt{37}}{4}$

5) $r = \frac{\pm 2\sqrt{21}}{7}$

6) $r = \frac{7 \pm \sqrt{209}}{16}$