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

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

Solve for x

$$x=5\sqrt{3}\approx 8.660254038$$

$$x=-5\sqrt{3}\approx -8.660254038$$

Steps by Finding Square Root

Calculate 5 to the power of 2 and get 25.

$$25+\left(5\sqrt{2}\right)^{2}=x^{2}$$

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

$$25+5^{2}\left(\sqrt{2}\right)^{2}=x^{2}$$

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{2} is 2.

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

Multiply 25 and 2 to get 50.

$$25+50=x^{2}$$

Add 25 and 50 to get 75.

$$75=x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}=75$$

Take the square root of both sides of the equation.

$$x=5\sqrt{3} x=-5\sqrt{3}$$

Steps Using the Quadratic Formula

Calculate 5 to the power of 2 and get 25.

$$25+\left(5\sqrt{2}\right)^{2}=x^{2}$$

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

$$25+5^{2}\left(\sqrt{2}\right)^{2}=x^{2}$$

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{2} is 2.

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

Multiply 25 and 2 to get 50.

$$25+50=x^{2}$$

Add 25 and 50 to get 75.

$$75=x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}=75$$

Subtract 75 from both sides.

$$x^{2}-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-75\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-75\right)}}{2}$$

Multiply -4 times -75.

$$x=\frac{0±\sqrt{300}}{2}$$

Take the square root of 300.

$$x=\frac{0±10\sqrt{3}}{2}$$

Now solve the equation x=\frac{0±10\sqrt{3}}{2} when ± is plus.

$$x=5\sqrt{3}$$

Now solve the equation x=\frac{0±10\sqrt{3}}{2} when ± is minus.

$$x=-5\sqrt{3}$$

The equation is now solved.

$$x=5\sqrt{3} x=-5\sqrt{3}$$