Answer :
### Solving the Given Quadratic Equations
Let's go through each of the given sets of quadratic equations and solve them step-by-step.
#### First Set of Equations
1. Equation: [tex]\(5 x^2 - 720 = 0\)[/tex]
[tex]\[ 5 x^2 = 720 \implies x^2 = \frac{720}{5} \implies x^2 = 144 \implies x = \pm \sqrt{144} \implies x = \pm 12 \][/tex]
Solution: [tex]\(\boxed{x = \pm 12}\)[/tex]
2. Equation: [tex]\(6 x^2 - 96 = 0\)[/tex]
[tex]\[ 6 x^2 = 96 \implies x^2 = \frac{96}{6} \implies x^2 = 16 \implies x = \pm \sqrt{16} \implies x = \pm 4 \][/tex]
Solution: [tex]\(\boxed{x = \pm 4}\)[/tex]
3. Equation: [tex]\(3 x^2 - 147 = 0\)[/tex]
[tex]\[ 3 x^2 = 147 \implies x^2 = \frac{147}{3} \implies x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
4. Equation: [tex]\(3 x^2 - 75 = 0\)[/tex]
[tex]\[ 3 x^2 = 75 \implies x^2 = \frac{75}{3} \implies x^2 = 25 \implies x = \pm \sqrt{25} \implies x = \pm 5 \][/tex]
Solution: [tex]\(\boxed{x = \pm 5}\)[/tex]
#### Second Set of Equations
1. Equation: [tex]\(x^2 - 9 = 0\)[/tex]
[tex]\[ x^2 = 9 \implies x = \pm \sqrt{9} \implies x = \pm 3 \][/tex]
Solution: [tex]\(\boxed{x = \pm 3}\)[/tex]
2. Equation: [tex]\(x^2 - 49 = 0\)[/tex]
[tex]\[ x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
3. Equation: [tex]\(3 x^2 - 1200 = 0\)[/tex]
[tex]\[ 3 x^2 = 1200 \implies x^2 = \frac{1200}{3} \implies x^2 = 400 \implies x = \pm \sqrt{400} \implies x = \pm 20 \][/tex]
Solution: [tex]\(\boxed{x = \pm 20}\)[/tex]
4. Equation: [tex]\(3 x^2 - 675 = 0\)[/tex]
[tex]\[ 3 x^2 = 675 \implies x^2 = \frac{675}{3} \implies x^2 = 225 \implies x = \pm \sqrt{225} \implies x = \pm 15 \][/tex]
Solution: [tex]\(\boxed{x = \pm 15}\)[/tex]
#### Third Set of Equations (Variations)
1. Equation: [tex]\(2 x^2 - 3 = 95\)[/tex]
[tex]\[ 2 x^2 - 3 = 95 \implies 2 x^2 = 98 \implies x^2 = \frac{98}{2} \implies x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
2. Equation: [tex]\(6 x^2 - 16 = 200\)[/tex]
[tex]\[ 6 x^2 - 16 = 200 \implies 6 x^2 = 216 \implies x^2 = \frac{216}{6} \implies x^2 = 36 \implies x = \pm \sqrt{36} \implies x = \pm 6 \][/tex]
Solution: [tex]\(\boxed{x = \pm 6}\)[/tex]
3. Equation: [tex]\(8 x^2 + 2 = 34\)[/tex]
[tex]\[ 8 x^2 + 2 = 34 \implies 8 x^2 = 32 \implies x^2 = \frac{32}{8} \implies x^2 = 4 \implies x = \pm \sqrt{4} \implies x = \pm 2 \][/tex]
Solution: [tex]\(\boxed{x = \pm 2}\)[/tex]
4. Equation: [tex]\(7 x^2 - 8 = 440\)[/tex]
[tex]\[ 7 x^2 - 8 = 440 \implies 7 x^2 = 448 \implies x^2 = \frac{448}{7} \implies x^2 = 64 \implies x = \pm \sqrt{64} \implies x = \pm 8 \][/tex]
Solution: [tex]\(\boxed{x = \pm 8}\)[/tex]
5. Equation: [tex]\(4 x^2 + 4 = 148\)[/tex]
[tex]\[ 4 x^2 + 4 = 148 \implies 4 x^2 = 144 \implies x^2 = \frac{144}{4} \implies x^2 = 36 \implies x = \pm \sqrt{36} \implies x = \pm 6 \][/tex]
Solution: [tex]\(\boxed{x = \pm 6}\)[/tex]
6. Equation: [tex]\(3 x^2 + 3 = 30\)[/tex]
[tex]\[ 3 x^2 + 3 = 30 \implies 3 x^2 = 27 \implies x^2 = \frac{27}{3} \implies x^2 = 9 \implies x = \pm \sqrt{9} \implies x = \pm 3 \][/tex]
Solution: [tex]\(\boxed{x = \pm 3}\)[/tex]
By solving each quadratic equation, we obtained the roots for each one. The solutions consist of the positive and negative square roots derived from isolating [tex]\(x^2\)[/tex] in each equation.
Let's go through each of the given sets of quadratic equations and solve them step-by-step.
#### First Set of Equations
1. Equation: [tex]\(5 x^2 - 720 = 0\)[/tex]
[tex]\[ 5 x^2 = 720 \implies x^2 = \frac{720}{5} \implies x^2 = 144 \implies x = \pm \sqrt{144} \implies x = \pm 12 \][/tex]
Solution: [tex]\(\boxed{x = \pm 12}\)[/tex]
2. Equation: [tex]\(6 x^2 - 96 = 0\)[/tex]
[tex]\[ 6 x^2 = 96 \implies x^2 = \frac{96}{6} \implies x^2 = 16 \implies x = \pm \sqrt{16} \implies x = \pm 4 \][/tex]
Solution: [tex]\(\boxed{x = \pm 4}\)[/tex]
3. Equation: [tex]\(3 x^2 - 147 = 0\)[/tex]
[tex]\[ 3 x^2 = 147 \implies x^2 = \frac{147}{3} \implies x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
4. Equation: [tex]\(3 x^2 - 75 = 0\)[/tex]
[tex]\[ 3 x^2 = 75 \implies x^2 = \frac{75}{3} \implies x^2 = 25 \implies x = \pm \sqrt{25} \implies x = \pm 5 \][/tex]
Solution: [tex]\(\boxed{x = \pm 5}\)[/tex]
#### Second Set of Equations
1. Equation: [tex]\(x^2 - 9 = 0\)[/tex]
[tex]\[ x^2 = 9 \implies x = \pm \sqrt{9} \implies x = \pm 3 \][/tex]
Solution: [tex]\(\boxed{x = \pm 3}\)[/tex]
2. Equation: [tex]\(x^2 - 49 = 0\)[/tex]
[tex]\[ x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
3. Equation: [tex]\(3 x^2 - 1200 = 0\)[/tex]
[tex]\[ 3 x^2 = 1200 \implies x^2 = \frac{1200}{3} \implies x^2 = 400 \implies x = \pm \sqrt{400} \implies x = \pm 20 \][/tex]
Solution: [tex]\(\boxed{x = \pm 20}\)[/tex]
4. Equation: [tex]\(3 x^2 - 675 = 0\)[/tex]
[tex]\[ 3 x^2 = 675 \implies x^2 = \frac{675}{3} \implies x^2 = 225 \implies x = \pm \sqrt{225} \implies x = \pm 15 \][/tex]
Solution: [tex]\(\boxed{x = \pm 15}\)[/tex]
#### Third Set of Equations (Variations)
1. Equation: [tex]\(2 x^2 - 3 = 95\)[/tex]
[tex]\[ 2 x^2 - 3 = 95 \implies 2 x^2 = 98 \implies x^2 = \frac{98}{2} \implies x^2 = 49 \implies x = \pm \sqrt{49} \implies x = \pm 7 \][/tex]
Solution: [tex]\(\boxed{x = \pm 7}\)[/tex]
2. Equation: [tex]\(6 x^2 - 16 = 200\)[/tex]
[tex]\[ 6 x^2 - 16 = 200 \implies 6 x^2 = 216 \implies x^2 = \frac{216}{6} \implies x^2 = 36 \implies x = \pm \sqrt{36} \implies x = \pm 6 \][/tex]
Solution: [tex]\(\boxed{x = \pm 6}\)[/tex]
3. Equation: [tex]\(8 x^2 + 2 = 34\)[/tex]
[tex]\[ 8 x^2 + 2 = 34 \implies 8 x^2 = 32 \implies x^2 = \frac{32}{8} \implies x^2 = 4 \implies x = \pm \sqrt{4} \implies x = \pm 2 \][/tex]
Solution: [tex]\(\boxed{x = \pm 2}\)[/tex]
4. Equation: [tex]\(7 x^2 - 8 = 440\)[/tex]
[tex]\[ 7 x^2 - 8 = 440 \implies 7 x^2 = 448 \implies x^2 = \frac{448}{7} \implies x^2 = 64 \implies x = \pm \sqrt{64} \implies x = \pm 8 \][/tex]
Solution: [tex]\(\boxed{x = \pm 8}\)[/tex]
5. Equation: [tex]\(4 x^2 + 4 = 148\)[/tex]
[tex]\[ 4 x^2 + 4 = 148 \implies 4 x^2 = 144 \implies x^2 = \frac{144}{4} \implies x^2 = 36 \implies x = \pm \sqrt{36} \implies x = \pm 6 \][/tex]
Solution: [tex]\(\boxed{x = \pm 6}\)[/tex]
6. Equation: [tex]\(3 x^2 + 3 = 30\)[/tex]
[tex]\[ 3 x^2 + 3 = 30 \implies 3 x^2 = 27 \implies x^2 = \frac{27}{3} \implies x^2 = 9 \implies x = \pm \sqrt{9} \implies x = \pm 3 \][/tex]
Solution: [tex]\(\boxed{x = \pm 3}\)[/tex]
By solving each quadratic equation, we obtained the roots for each one. The solutions consist of the positive and negative square roots derived from isolating [tex]\(x^2\)[/tex] in each equation.
