Question

1. $7x^{2}+66 = 3$
2. $\frac{1}{2}x^{2}+3 = 75$
3. $9x^{2}-16 = 0$
4. $25x^{2}+10 = 46$
5. $5x^{2}-1 = x^{2}$
6. $16x^{2}-34 = 15$

Explanation:

Problem 11: $7x^2 + 66 = 3$

Step1: Isolate $x^2$ term

Subtract 66 from both sides:
$7x^2 = 3 - 66$
$7x^2 = -63$

Step2: Solve for $x^2$

Divide by 7:
$x^2 = \frac{-63}{7} = -9$

Step3: Solve for $x$

Take square root of negative number:
$x = \pm 3i$

---

Problem 12: $\frac{1}{2}x^2 + 3 = 75$

Step1: Isolate $x^2$ term

Subtract 3 from both sides:
$\frac{1}{2}x^2 = 75 - 3$
$\frac{1}{2}x^2 = 72$

Step2: Solve for $x^2$

Multiply by 2:
$x^2 = 72 \times 2 = 144$

Step3: Solve for $x$

Take square root:
$x = \pm \sqrt{144} = \pm 12$

---

Problem 13: $9x^2 - 16 = 0$

Step1: Isolate $x^2$ term

Add 16 to both sides:
$9x^2 = 16$

Step2: Solve for $x^2$

Divide by 9:
$x^2 = \frac{16}{9}$

Step3: Solve for $x$

Take square root:
$x = \pm \sqrt{\frac{16}{9}} = \pm \frac{4}{3}$

---

Problem 14: $25x^2 + 10 = 46$

Step1: Isolate $x^2$ term

Subtract 10 from both sides:
$25x^2 = 46 - 10$
$25x^2 = 36$

Step2: Solve for $x^2$

Divide by 25:
$x^2 = \frac{36}{25}$

Step3: Solve for $x$

Take square root:
$x = \pm \sqrt{\frac{36}{25}} = \pm \frac{6}{5}$

---

Problem 15: $5x^2 - 1 = x^2$

Step1: Combine like terms

Subtract $x^2$ from both sides:
$5x^2 - x^2 = 1$
$4x^2 = 1$

Step2: Solve for $x^2$

Divide by 4:
$x^2 = \frac{1}{4}$

Step3: Solve for $x$

Take square root:
$x = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2}$

---

Problem 16: $16x^2 - 34 = 15$

Step1: Isolate $x^2$ term

Add 34 to both sides:
$16x^2 = 15 + 34$
$16x^2 = 49$

Step2: Solve for $x^2$

Divide by 16:
$x^2 = \frac{49}{16}$

Step3: Solve for $x$

Take square root:
$x = \pm \sqrt{\frac{49}{16}} = \pm \frac{7}{4}$

Answer:

1. $x = 3i$ and $x = -3i$
2. $x = 12$ and $x = -12$
3. $x = \frac{4}{3}$ and $x = -\frac{4}{3}$
4. $x = \frac{6}{5}$ and $x = -\frac{6}{5}$
5. $x = \frac{1}{2}$ and $x = -\frac{1}{2}$
6. $x = \frac{7}{4}$ and $x = -\frac{7}{4}$