Solve frac{sqrt{3left(x+hright)^2+5}-sqrt{3x^2+5}}{h}=0

$$\frac{ \sqrt{ 3 { \left(x+h \right) }^{ 2 } +5 } - \sqrt{ 3 { x }^{ 2 } +5 } }{ h } =0$$

Solve for x

$$x=-\frac{h}{2}$$

$$h\neq 0$$

Solution Steps

Multiply both sides of the equation by h.

$$\sqrt{3\left(x+h\right)^{2}+5}-\sqrt{3x^{2}+5}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+h\right)^{2}.

$$\sqrt{3\left(x^{2}+2xh+h^{2}\right)+5}-\sqrt{3x^{2}+5}=0$$

Use the distributive property to multiply 3 by x^{2}+2xh+h^{2}.

$$\sqrt{3x^{2}+6xh+3h^{2}+5}-\sqrt{3x^{2}+5}=0$$

Subtract -\sqrt{3x^{2}+5} from both sides of the equation.

$$\sqrt{3x^{2}+6xh+3h^{2}+5}=\sqrt{3x^{2}+5}$$

Square both sides of the equation.

$$\left(\sqrt{3x^{2}+6xh+3h^{2}+5}\right)^{2}=\left(\sqrt{3x^{2}+5}\right)^{2}$$

Calculate \sqrt{3x^{2}+6xh+3h^{2}+5} to the power of 2 and get 3x^{2}+6xh+3h^{2}+5.

$$3x^{2}+6xh+3h^{2}+5=\left(\sqrt{3x^{2}+5}\right)^{2}$$

Calculate \sqrt{3x^{2}+5} to the power of 2 and get 3x^{2}+5.

$$3x^{2}+6xh+3h^{2}+5=3x^{2}+5$$

Subtract 3x^{2} from both sides.

$$3x^{2}+6xh+3h^{2}+5-3x^{2}=5$$

Combine 3x^{2} and -3x^{2} to get 0.

$$6xh+3h^{2}+5=5$$

Subtract 3h^{2} from both sides.

$$6xh+5=5-3h^{2}$$

Subtract 5 from both sides.

$$6xh=5-3h^{2}-5$$

Subtract 5 from 5 to get 0.

$$6xh=-3h^{2}$$

The equation is in standard form.

$$6hx=-3h^{2}$$

Divide both sides by 6h.

$$\frac{6hx}{6h}=-\frac{3h^{2}}{6h}$$

Dividing by 6h undoes the multiplication by 6h.

$$x=-\frac{3h^{2}}{6h}$$

Divide -3h^{2} by 6h.

$$x=-\frac{h}{2}$$

Substitute -\frac{h}{2} for x in the equation \frac{\sqrt{3\left(x+h\right)^{2}+5}-\sqrt{3x^{2}+5}}{h}=0.

$$\frac{\sqrt{3\left(-\frac{h}{2}+h\right)^{2}+5}-\sqrt{3\left(-\frac{h}{2}\right)^{2}+5}}{h}=0$$

Simplify. The value x=-\frac{h}{2} satisfies the equation.

$$0=0$$

Equation \sqrt{3x^{2}+6hx+3h^{2}+5}=\sqrt{3x^{2}+5} has a unique solution.

$$x=-\frac{h}{2}$$

Solve for h

$$h=-2x$$

$$x\neq 0$$