We are asked to solve the given equations for their respective variables. The equations are: a) $x^2 + 2 = 18$ b) $y^2 - 5 = 22$ c) $9 - x^2 = 5$ d) $7 - 4t^2 = 18t^2$ e) $12x^2 - 7 = -7$ f) $2.25a^2 + 15 = 12$ g) $2t + 8 = t(2 + 3t)$ h) $x(2x - 6) = 4 - 6x$ i) $z(z^2 + 8) = 0$ j) $7x(3x + 5) - 11 = 5x(2x + 7)$ k) $(x + 3)^2 - 7x + 8 = (x - 5)(x + 5) - x$

1. Problem Description

We are asked to solve the given equations for their respective variables. The equations are:

a) $x^2 + 2 = 18$

b) $y^2 - 5 = 22$

c) $9 - x^2 = 5$

d) $7 - 4t^2 = 18t^2$

e) $12x^2 - 7 = -7$

f) $2.25a^2 + 15 = 12$

g) $2t + 8 = t(2 + 3t)$

h) $x(2x - 6) = 4 - 6x$

i) $z(z^2 + 8) = 0$

j) $7x(3x + 5) - 11 = 5x(2x + 7)$

k) $(x + 3)^2 - 7x + 8 = (x - 5)(x + 5) - x$

2. Solution Steps

a) $x^2 + 2 = 18$

$x^2 = 18 - 2$

$x^2 = 16$

$x = \pm \sqrt{16}$

$x = \pm 4$

b) $y^2 - 5 = 22$

$y^2 = 22 + 5$

$y^2 = 27$

$y = \pm \sqrt{27}$

$y = \pm 3\sqrt{3}$

c) $9 - x^2 = 5$

$9 - 5 = x^2$

$x^2 = 4$

$x = \pm \sqrt{4}$

$x = \pm 2$

d) $7 - 4t^2 = 18t^2$

$7 = 18t^2 + 4t^2$

$7 = 22t^2$

$t^2 = \frac{7}{22}$

$t = \pm \sqrt{\frac{7}{22}}$

$t = \pm \frac{\sqrt{154}}{22}$

e) $12x^2 - 7 = -7$

$12x^2 = -7 + 7$

$12x^2 = 0$

$x^2 = 0$

$x = 0$

f) $2.25a^2 + 15 = 12$

$2.25a^2 = 12 - 15$

$2.25a^2 = -3$

$a^2 = \frac{-3}{2.25} = -\frac{4}{3}$

Since $a^2$ cannot be negative for real numbers, there is no real solution.

g) $2t + 8 = t(2 + 3t)$

$2t + 8 = 2t + 3t^2$

$3t^2 = 8$

$t^2 = \frac{8}{3}$

$t = \pm \sqrt{\frac{8}{3}}$

$t = \pm \frac{2\sqrt{2}}{\sqrt{3}} = \pm \frac{2\sqrt{6}}{3}$

h) $x(2x - 6) = 4 - 6x$

$2x^2 - 6x = 4 - 6x$

$2x^2 - 6x + 6x - 4 = 0$

$2x^2 - 4 = 0$

$2x^2 = 4$

$x^2 = 2$

$x = \pm \sqrt{2}$

i) $z(z^2 + 8) = 0$

So either $z = 0$ or $z^2 + 8 = 0$.

$z^2 = -8$, which means $z = \pm \sqrt{-8} = \pm 2i\sqrt{2}$.

If we only consider real solutions, then $z = 0$.

j) $7x(3x + 5) - 11 = 5x(2x + 7)$

$21x^2 + 35x - 11 = 10x^2 + 35x$

$21x^2 - 10x^2 + 35x - 35x - 11 = 0$

$11x^2 - 11 = 0$

$11x^2 = 11$

$x^2 = 1$

$x = \pm 1$

k) $(x + 3)^2 - 7x + 8 = (x - 5)(x + 5) - x$

$x^2 + 6x + 9 - 7x + 8 = x^2 - 25 - x$

$x^2 - x + 17 = x^2 - x - 25$

$17 = -25$

This is a contradiction, so there is no solution.

3. Final Answer

a) $x = \pm 4$

b) $y = \pm 3\sqrt{3}$

c) $x = \pm 2$

d) $t = \pm \frac{\sqrt{154}}{22}$

e) $x = 0$

f) No real solution

g) $t = \pm \frac{2\sqrt{6}}{3}$

h) $x = \pm \sqrt{2}$

i) $z = 0$ (real solution)

j) $x = \pm 1$

k) No solution