Solve (5x)^2=(2x)^2+(2sqrt{2})^2

$$( 5 x ) ^ { 2 } = ( 2 x ) ^ { 2 } + ( 2 \sqrt { 2 } ) ^ { 2 }$$

Solve for x

$$x=\frac{2\sqrt{42}}{21}\approx 0.6172134$$

$$x=-\frac{2\sqrt{42}}{21}\approx -0.6172134$$

Steps by Finding Square Root

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(2x\right)^{2}+\left(2\sqrt{2}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(2x\right)^{2}+\left(2\sqrt{2}\right)^{2}$$

Expand \left(2x\right)^{2}.

$$25x^{2}=2^{2}x^{2}+\left(2\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}=4x^{2}+\left(2\sqrt{2}\right)^{2}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$25x^{2}=4x^{2}+2^{2}\left(\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}=4x^{2}+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$25x^{2}=4x^{2}+4\times 2$$

Multiply 4 and 2 to get 8.

$$25x^{2}=4x^{2}+8$$

Subtract 4x^{2} from both sides.

$$25x^{2}-4x^{2}=8$$

Combine 25x^{2} and -4x^{2} to get 21x^{2}.

$$21x^{2}=8$$

Divide both sides by 21.

$$x^{2}=\frac{8}{21}$$

Take the square root of both sides of the equation.

$$x=\frac{2\sqrt{42}}{21} x=-\frac{2\sqrt{42}}{21}$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(2x\right)^{2}+\left(2\sqrt{2}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(2x\right)^{2}+\left(2\sqrt{2}\right)^{2}$$

Expand \left(2x\right)^{2}.

$$25x^{2}=2^{2}x^{2}+\left(2\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}=4x^{2}+\left(2\sqrt{2}\right)^{2}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$25x^{2}=4x^{2}+2^{2}\left(\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}=4x^{2}+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$25x^{2}=4x^{2}+4\times 2$$

Multiply 4 and 2 to get 8.

$$25x^{2}=4x^{2}+8$$

Subtract 4x^{2} from both sides.

$$25x^{2}-4x^{2}=8$$

Combine 25x^{2} and -4x^{2} to get 21x^{2}.

$$21x^{2}=8$$

Subtract 8 from both sides.

$$21x^{2}-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, 0 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 21\left(-8\right)}}{2\times 21}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 21\left(-8\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{0±\sqrt{-84\left(-8\right)}}{2\times 21}$$

Multiply -84 times -8.

$$x=\frac{0±\sqrt{672}}{2\times 21}$$

Take the square root of 672.

$$x=\frac{0±4\sqrt{42}}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{0±4\sqrt{42}}{42}$$

Now solve the equation x=\frac{0±4\sqrt{42}}{42} when ± is plus.

$$x=\frac{2\sqrt{42}}{21}$$

Now solve the equation x=\frac{0±4\sqrt{42}}{42} when ± is minus.

$$x=-\frac{2\sqrt{42}}{21}$$

The equation is now solved.

$$x=\frac{2\sqrt{42}}{21} x=-\frac{2\sqrt{42}}{21}$$