Solve (5x)^2+(3x)^2=8^2

$$( 5 x ) ^ { 2 } + ( 3 x ) ^ { 2 } = 8 ^ { 2 }$$

Solve for x

$$x = \frac{4 \sqrt{34}}{17} \approx 1.371988681$$

$$x = -\frac{4 \sqrt{34}}{17} \approx -1.371988681$$

Steps by Finding Square Root

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

$$5^{2}x^{2}+\left(3x\right)^{2}=8^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+\left(3x\right)^{2}=8^{2}$$

Expand \left(3x\right)^{2}.

$$25x^{2}+3^{2}x^{2}=8^{2}$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}+9x^{2}=8^{2}$$

Combine 25x^{2} and 9x^{2} to get 34x^{2}.

$$34x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$34x^{2}=64$$

Divide both sides by 34.

$$x^{2}=\frac{64}{34}$$

Reduce the fraction \frac{64}{34} to lowest terms by extracting and canceling out 2.

$$x^{2}=\frac{32}{17}$$

Take the square root of both sides of the equation.

$$x=\frac{4\sqrt{34}}{17} x=-\frac{4\sqrt{34}}{17}$$

Steps Using the Quadratic Formula

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

$$5^{2}x^{2}+\left(3x\right)^{2}=8^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+\left(3x\right)^{2}=8^{2}$$

Expand \left(3x\right)^{2}.

$$25x^{2}+3^{2}x^{2}=8^{2}$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}+9x^{2}=8^{2}$$

Combine 25x^{2} and 9x^{2} to get 34x^{2}.

$$34x^{2}=8^{2}$$

Calculate 8 to the power of 2 and get 64.

$$34x^{2}=64$$

Subtract 64 from both sides.

$$34x^{2}-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 34 for a, 0 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 34\left(-64\right)}}{2\times 34}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 34\left(-64\right)}}{2\times 34}$$

Multiply -4 times 34.

$$x=\frac{0±\sqrt{-136\left(-64\right)}}{2\times 34}$$

Multiply -136 times -64.

$$x=\frac{0±\sqrt{8704}}{2\times 34}$$

Take the square root of 8704.

$$x=\frac{0±16\sqrt{34}}{2\times 34}$$

Multiply 2 times 34.

$$x=\frac{0±16\sqrt{34}}{68}$$

Now solve the equation x=\frac{0±16\sqrt{34}}{68} when ± is plus.

$$x=\frac{4\sqrt{34}}{17}$$

Now solve the equation x=\frac{0±16\sqrt{34}}{68} when ± is minus.

$$x=-\frac{4\sqrt{34}}{17}$$

The equation is now solved.

$$x=\frac{4\sqrt{34}}{17} x=-\frac{4\sqrt{34}}{17}$$