Solve 105^2=(9x)^2+(32x)^2

$$105 ^ { 2 } = ( 9 x ) ^ { 2 } + ( 32 x ) ^ { 2 }$$

Solve for x

$$x = \frac{21 \sqrt{1105}}{221} \approx 3.158698397$$

$$x = -\frac{21 \sqrt{1105}}{221} \approx -3.158698397$$

Steps by Finding Square Root

Calculate 105 to the power of 2 and get 11025.

$$11025=\left(9x\right)^{2}+\left(32x\right)^{2}$$

Expand \left(9x\right)^{2}.

$$11025=9^{2}x^{2}+\left(32x\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$11025=81x^{2}+\left(32x\right)^{2}$$

Expand \left(32x\right)^{2}.

$$11025=81x^{2}+32^{2}x^{2}$$

Calculate 32 to the power of 2 and get 1024.

$$11025=81x^{2}+1024x^{2}$$

Combine 81x^{2} and 1024x^{2} to get 1105x^{2}.

$$11025=1105x^{2}$$

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

$$1105x^{2}=11025$$

Divide both sides by 1105.

$$x^{2}=\frac{11025}{1105}$$

Reduce the fraction \frac{11025}{1105} to lowest terms by extracting and canceling out 5.

$$x^{2}=\frac{2205}{221}$$

Take the square root of both sides of the equation.

$$x=\frac{21\sqrt{1105}}{221} x=-\frac{21\sqrt{1105}}{221}$$

Steps Using the Quadratic Formula

Calculate 105 to the power of 2 and get 11025.

$$11025=\left(9x\right)^{2}+\left(32x\right)^{2}$$

Expand \left(9x\right)^{2}.

$$11025=9^{2}x^{2}+\left(32x\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$11025=81x^{2}+\left(32x\right)^{2}$$

Expand \left(32x\right)^{2}.

$$11025=81x^{2}+32^{2}x^{2}$$

Calculate 32 to the power of 2 and get 1024.

$$11025=81x^{2}+1024x^{2}$$

Combine 81x^{2} and 1024x^{2} to get 1105x^{2}.

$$11025=1105x^{2}$$

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

$$1105x^{2}=11025$$

Subtract 11025 from both sides.

$$1105x^{2}-11025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1105 for a, 0 for b, and -11025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 1105\left(-11025\right)}}{2\times 1105}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 1105\left(-11025\right)}}{2\times 1105}$$

Multiply -4 times 1105.

$$x=\frac{0±\sqrt{-4420\left(-11025\right)}}{2\times 1105}$$

Multiply -4420 times -11025.

$$x=\frac{0±\sqrt{48730500}}{2\times 1105}$$

Take the square root of 48730500.

$$x=\frac{0±210\sqrt{1105}}{2\times 1105}$$

Multiply 2 times 1105.

$$x=\frac{0±210\sqrt{1105}}{2210}$$

Now solve the equation x=\frac{0±210\sqrt{1105}}{2210} when ± is plus.

$$x=\frac{21\sqrt{1105}}{221}$$

Now solve the equation x=\frac{0±210\sqrt{1105}}{2210} when ± is minus.

$$x=-\frac{21\sqrt{1105}}{221}$$

The equation is now solved.

$$x=\frac{21\sqrt{1105}}{221} x=-\frac{21\sqrt{1105}}{221}$$