Question

1. $10-2x^{2}=-198$
2. $\frac{1}{4}x^{2}-7=25$
3. $(x-4)^{2}=9$
4. $(x+1)^{2}=75$
5. $2(x+3)^{2}=56$
6. $\frac{3}{2}(x-2)^{2}=72$

Explanation:

Problem 11:

Step1: Isolate the $x^2$ term

$10 - 2x^2 = -198 \\
-2x^2 = -198 - 10 \\
-2x^2 = -208$

Step2: Solve for $x^2$

$x^2 = \frac{-208}{-2} \\
x^2 = 104$

Step3: Find square roots

$x = \pm\sqrt{104} = \pm2\sqrt{26}$

Problem 12:

Step1: Isolate the $x^2$ term

$\frac{1}{4}x^2 - 7 = 25 \\
\frac{1}{4}x^2 = 25 + 7 \\
\frac{1}{4}x^2 = 32$

Step2: Solve for $x^2$

$x^2 = 32 \times 4 \\
x^2 = 128$

Step3: Find square roots

$x = \pm\sqrt{128} = \pm8\sqrt{2}$

Problem 13:

Step1: Take square roots directly

$(x-4)^2 = 9 \\
x-4 = \pm\sqrt{9} = \pm3$

Step2: Solve for $x$

$x = 4 + 3 = 7 \text{ or } x = 4 - 3 = 1$

Problem 14:

Step1: Take square roots directly

$(x+1)^2 = 75 \\
x+1 = \pm\sqrt{75} = \pm5\sqrt{3}$

Step2: Solve for $x$

$x = -1 + 5\sqrt{3} \text{ or } x = -1 - 5\sqrt{3}$

Problem 15:

Step1: Isolate the squared term

$2(x+3)^2 = 56 \\
(x+3)^2 = \frac{56}{2} \\
(x+3)^2 = 28$

Step2: Take square roots

$x+3 = \pm\sqrt{28} = \pm2\sqrt{7}$

Step3: Solve for $x$

$x = -3 + 2\sqrt{7} \text{ or } x = -3 - 2\sqrt{7}$

Problem 16:

Step1: Isolate the squared term

$\frac{3}{2}(x-2)^2 = 72 \\
(x-2)^2 = 72 \times \frac{2}{3} \\
(x-2)^2 = 48$

Step2: Take square roots

$x-2 = \pm\sqrt{48} = \pm4\sqrt{3}$

Step3: Solve for $x$

$x = 2 + 4\sqrt{3} \text{ or } x = 2 - 4\sqrt{3}$

Answer:

1. $x = 2\sqrt{26}$ or $x = -2\sqrt{26}$
2. $x = 8\sqrt{2}$ or $x = -8\sqrt{2}$
3. $x = 7$ or $x = 1$
4. $x = -1 + 5\sqrt{3}$ or $x = -1 - 5\sqrt{3}$
5. $x = -3 + 2\sqrt{7}$ or $x = -3 - 2\sqrt{7}$
6. $x = 2 + 4\sqrt{3}$ or $x = 2 - 4\sqrt{3}$