Solve sqrt{15x^2}=sqrt{4-left(2-xright)^2}

$$\sqrt{ 15 { x }^{ 2 } } = \sqrt{ 4- { \left(2-x \right) }^{ 2 } }$$

Solve for x

$$x=\frac{1}{4}=0.25$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{15x^{2}}\right)^{2}=\left(\sqrt{4-\left(2-x\right)^{2}}\right)^{2}$$

Calculate \sqrt{15x^{2}} to the power of 2 and get 15x^{2}.

$$15x^{2}=\left(\sqrt{4-\left(2-x\right)^{2}}\right)^{2}$$

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

$$15x^{2}=\left(\sqrt{4-\left(4-4x+x^{2}\right)}\right)^{2}$$

To find the opposite of 4-4x+x^{2}, find the opposite of each term.

$$15x^{2}=\left(\sqrt{4-4+4x-x^{2}}\right)^{2}$$

Subtract 4 from 4 to get 0.

$$15x^{2}=\left(\sqrt{4x-x^{2}}\right)^{2}$$

Calculate \sqrt{4x-x^{2}} to the power of 2 and get 4x-x^{2}.

$$15x^{2}=4x-x^{2}$$

Subtract 4x from both sides.

$$15x^{2}-4x=-x^{2}$$

Add x^{2} to both sides.

$$15x^{2}-4x+x^{2}=0$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-4x=0$$

Factor out x.

$$x\left(16x-4\right)=0$$

To find equation solutions, solve x=0 and 16x-4=0.

$$x=0 x=\frac{1}{4}$$

Substitute 0 for x in the equation \sqrt{15x^{2}}=\sqrt{4-\left(2-x\right)^{2}}.

$$\sqrt{15\times 0^{2}}=\sqrt{4-\left(2-0\right)^{2}}$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Substitute \frac{1}{4} for x in the equation \sqrt{15x^{2}}=\sqrt{4-\left(2-x\right)^{2}}.

$$\sqrt{15\times \left(\frac{1}{4}\right)^{2}}=\sqrt{4-\left(2-\frac{1}{4}\right)^{2}}$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$\frac{1}{4}\times 15^{\frac{1}{2}}=\frac{1}{4}\times 15^{\frac{1}{2}}$$

List all solutions of \sqrt{15x^{2}}=\sqrt{4-\left(2-x\right)^{2}}.

$$x=0 x=\frac{1}{4}$$