Question

zeros can be irrational! solve the following quadratic equations. write all answers in simplest radical form. 17. $x^2 - 3 = 0$ 18. $x^2 - 8 = 0$ 19. $x^2 + 8 = 56$ 20. $x^2 - 16 = 59$ 21. $2x^2 - 126 = 0$ 22. $8 - 3x^2 = -22$ 23. $-x^2 - 10 = 18$ 24. $\frac{1}{2}x^2 - 11 = 25$

Explanation:

Problem 17: $x^2 - 3 = 0$

Step1: Isolate $x^2$

$x^2 = 3$

Step2: Take square root of both sides

$x = \pm\sqrt{3}$

Problem 18: $x^2 - 8 = 0$

Step1: Isolate $x^2$

$x^2 = 8$

Step2: Simplify radical and solve

$x = \pm\sqrt{8} = \pm2\sqrt{2}$

Problem 19: $x^2 + 8 = 56$

Step1: Isolate $x^2$

$x^2 = 56 - 8 = 48$

Step2: Simplify radical and solve

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

Problem 20: $x^2 - 16 = 59$

Step1: Isolate $x^2$

$x^2 = 59 + 16 = 75$

Step2: Simplify radical and solve

$x = \pm\sqrt{75} = \pm5\sqrt{3}$

Problem 21: $2x^2 - 126 = 0$

Step1: Isolate $x^2$

$2x^2 = 126 \implies x^2 = 63$

Step2: Simplify radical and solve

$x = \pm\sqrt{63} = \pm3\sqrt{7}$

Problem 22: $8 - 3x^2 = -22$

Step1: Isolate $x^2$

$-3x^2 = -22 - 8 = -30 \implies x^2 = 10$

Step2: Solve for x

$x = \pm\sqrt{10}$

Problem 23: $-x^2 - 10 = 18$

Step1: Isolate $x^2$

$-x^2 = 18 + 10 = 28 \implies x^2 = -28$

Step2: Identify no real solutions

Square of real number cannot be negative.

Problem 24: $\frac{1}{2}x^2 - 11 = 25$

Step1: Isolate $x^2$

$\frac{1}{2}x^2 = 25 + 11 = 36 \implies x^2 = 72$

Step2: Simplify radical and solve

$x = \pm\sqrt{72} = \pm6\sqrt{2}$

Answer:

1. $x = \sqrt{3}$ and $x = -\sqrt{3}$
2. $x = 2\sqrt{2}$ and $x = -2\sqrt{2}$
3. $x = 4\sqrt{3}$ and $x = -4\sqrt{3}$
4. $x = 5\sqrt{3}$ and $x = -5\sqrt{3}$
5. $x = 3\sqrt{7}$ and $x = -3\sqrt{7}$
6. $x = \sqrt{10}$ and $x = -\sqrt{10}$
7. No real solutions
8. $x = 6\sqrt{2}$ and $x = -6\sqrt{2}$