Solve x^2+(16x/9)^2=32^2

$$x ^ { 2 } + ( \frac { 16 x } { 9 } ) ^ { 2 } = 32 ^ { 2 }$$

Solve for x

$$x = \frac{288 \sqrt{337}}{337} \approx 15.688359668$$

$$x = -\frac{288 \sqrt{337}}{337} \approx -15.688359668$$

Steps by Finding Square Root

To raise \frac{16x}{9} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{\left(16x\right)^{2}}{9^{2}}=32^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{9^{2}}{9^{2}}.

$$\frac{x^{2}\times 9^{2}}{9^{2}}+\frac{\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Since \frac{x^{2}\times 9^{2}}{9^{2}} and \frac{\left(16x\right)^{2}}{9^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}\times 9^{2}+\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Do the multiplications in x^{2}\times 9^{2}+\left(16x\right)^{2}.

$$\frac{81x^{2}+\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Combine like terms in 81x^{2}+\left(16x\right)^{2}.

$$\frac{337x^{2}}{9^{2}}=32^{2}$$

Calculate 32 to the power of 2 and get 1024.

$$\frac{337x^{2}}{9^{2}}=1024$$

Calculate 9 to the power of 2 and get 81.

$$\frac{337x^{2}}{81}=1024$$

Multiply both sides by 81.

$$337x^{2}=1024\times 81$$

Multiply 1024 and 81 to get 82944.

$$337x^{2}=82944$$

Divide both sides by 337.

$$x^{2}=\frac{82944}{337}$$

Take the square root of both sides of the equation.

$$x=\frac{288\sqrt{337}}{337} x=-\frac{288\sqrt{337}}{337}$$

Steps Using the Quadratic Formula

To raise \frac{16x}{9} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{\left(16x\right)^{2}}{9^{2}}=32^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{9^{2}}{9^{2}}.

$$\frac{x^{2}\times 9^{2}}{9^{2}}+\frac{\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Since \frac{x^{2}\times 9^{2}}{9^{2}} and \frac{\left(16x\right)^{2}}{9^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}\times 9^{2}+\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Do the multiplications in x^{2}\times 9^{2}+\left(16x\right)^{2}.

$$\frac{81x^{2}+\left(16x\right)^{2}}{9^{2}}=32^{2}$$

Combine like terms in 81x^{2}+\left(16x\right)^{2}.

$$\frac{337x^{2}}{9^{2}}=32^{2}$$

Calculate 32 to the power of 2 and get 1024.

$$\frac{337x^{2}}{9^{2}}=1024$$

Calculate 9 to the power of 2 and get 81.

$$\frac{337x^{2}}{81}=1024$$

Subtract 1024 from both sides.

$$\frac{337x^{2}}{81}-1024=0$$

Multiply both sides of the equation by 81.

$$337x^{2}-82944=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 337 for a, 0 for b, and -82944 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 337\left(-82944\right)}}{2\times 337}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 337\left(-82944\right)}}{2\times 337}$$

Multiply -4 times 337.

$$x=\frac{0±\sqrt{-1348\left(-82944\right)}}{2\times 337}$$

Multiply -1348 times -82944.

$$x=\frac{0±\sqrt{111808512}}{2\times 337}$$

Take the square root of 111808512.

$$x=\frac{0±576\sqrt{337}}{2\times 337}$$

Multiply 2 times 337.

$$x=\frac{0±576\sqrt{337}}{674}$$

Now solve the equation x=\frac{0±576\sqrt{337}}{674} when ± is plus.

$$x=\frac{288\sqrt{337}}{337}$$

Now solve the equation x=\frac{0±576\sqrt{337}}{674} when ± is minus.

$$x=-\frac{288\sqrt{337}}{337}$$

The equation is now solved.

$$x=\frac{288\sqrt{337}}{337} x=-\frac{288\sqrt{337}}{337}$$