Solve the following radical equations a) sqrt(4-x) - sqrt(6+x) = sqrt(14+2x) b) 2sqrt(2x-12) + sqrt(2x-3) - 3 = 0

$$(\sqrt{4-x})^2 = (\sqrt{14+2x} + \sqrt{6+x})^2$$ $$4-x = (14+2x) + 2\sqrt{(14+2x)(6+x)} + (6+x)$$ $$4-x = 20+3x + 2\sqrt{84+14x+12x+2x^2}$$ $$4-x = 20+3x + 2\sqrt{2x^2+26x+84}$$

$$4-x - (20+3x) = 2\sqrt{2x^2+26x+84}$$ $$-16-4x = 2\sqrt{2x^2+26x+84}$$

Gawanya pande zote mbili kwa 2:

$$-8-2x = \sqrt{2x^2+26x+84}$$

$$(-8-2x)^2 = (\sqrt{2x^2+26x+84})^2$$ $$(2x+8)^2 = 2x^2+26x+84$$ $$4x^2+32x+64 = 2x^2+26x+84$$

$$4x^2-2x^2+32x-26x+64-84 = 0$$ $$2x^2+6x-20 = 0$$

Gawanya kwa 2:

$$x^2+3x-10 = 0$$

$$(x+5)(x-2) = 0$$

Hivyo, $x=-5$ au $x=2$.
Angalia majibu katika mlinganyo asili.
Kwa $x=-5$: $\sqrt{4-(-5)} - \sqrt{6+(-5)} = \sqrt{9} - \sqrt{1} = 3-1=2$. Na $\sqrt{14+2(-5)} = \sqrt{14-10} = \sqrt{4}=2$. Hivyo $x=-5$ ni suluhisho sahihi.
Kwa $x=2$: $\sqrt{4-2} - \sqrt{6+2} = \sqrt{2} - \sqrt{8} = \sqrt{2} - 2\sqrt{2} = -\sqrt{2}$. Na $\sqrt{14+2(2)} = \sqrt{18} = 3\sqrt{2}$. Kwa kuwa $-\sqrt{2} \neq 3\sqrt{2}$, $x=2$ si suluhisho sahihi.

Suluhisho la mlinganyo ni:

$$\boxed{x=-5}$$

$$(2\sqrt{2x-12})^2 = (3 - \sqrt{2x-3})^2$$ $$4(2x-12) = 3^2 - 2(3)\sqrt{2x-3} + (\sqrt{2x-3})^2$$ $$8x-48 = 9 - 6\sqrt{2x-3} + 2x-3$$ $$8x-48 = 6+2x - 6\sqrt{2x-3}$$

$$8x-48 - (6+2x) = -6\sqrt{2x-3}$$ $$6x-54 = -6\sqrt{2x-3}$$

$$\frac{6x-54}{-6} = \sqrt{2x-3}$$ $$-x+9 = \sqrt{2x-3}$$

$$(-x+9)^2 = (\sqrt{2x-3})^2$$ $$x^2-18x+81 = 2x-3$$

$$x^2-18x-2x+81+3 = 0$$ $$x^2-20x+84 = 0$$

$$x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(84)}}{2(1)}$$ $$x = \frac{20 \pm \sqrt{400 - 336}}{2}$$ $$x = \frac{20 \pm \sqrt{64}}{2}$$ $$x = \frac{20 \pm 8}{2}$$

Hivyo, $x_1 = \frac{20+8}{2} = \frac{28}{2} = 14$ au $x_2 = \frac{20-8}{2} = \frac{12}{2} = 6$.

Kwa $x=14$: $2\sqrt{2(14)-12} + \sqrt{2(14)-3} - 3 = 2\sqrt{16} + \sqrt{25} - 3 = 2(4) + 5 - 3 = 8+5-3 = 10$. Kwa kuwa $10 \neq 0$, $x=14$ si suluhisho sahihi.
Kwa $x=6$: $2\sqrt{2(6)-12} + \sqrt{2(6)-3} - 3 = 2\sqrt{0} + \sqrt{9} - 3 = 2(0) + 3 - 3 = 0$. Hivyo $x=6$ ni suluhisho sahihi.

Suluhisho la mlinganyo ni:

$$\boxed{x=6}$$