Solve 13^2-{left(1/2xright)}^2={left(120/xright)}^2

$${ 13 }^{ 2 } - { \left( \frac{ 1 }{ 2 } x \right) }^{ 2 } = { \left( \frac{ 120 }{ x } \right) }^{ 2 }$$

Solve for x

$$x=10$$

$$x=24$$

$$x=-10$$

$$x=-24$$

Steps Using the Quadratic Formula

Calculate 13 to the power of 2 and get 169.

$$169-\left(\frac{1}{2}x\right)^{2}=\left(\frac{120}{x}\right)^{2}$$

Expand \left(\frac{1}{2}x\right)^{2}.

$$169-\left(\frac{1}{2}\right)^{2}x^{2}=\left(\frac{120}{x}\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$169-\frac{1}{4}x^{2}=\left(\frac{120}{x}\right)^{2}$$

To raise \frac{120}{x} to a power, raise both numerator and denominator to the power and then divide.

$$169-\frac{1}{4}x^{2}=\frac{120^{2}}{x^{2}}$$

Calculate 120 to the power of 2 and get 14400.

$$169-\frac{1}{4}x^{2}=\frac{14400}{x^{2}}$$

Subtract \frac{14400}{x^{2}} from both sides.

$$169-\frac{1}{4}x^{2}-\frac{14400}{x^{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of 4,x^{2}.

$$4x^{2}\times 169-\frac{1}{4}x^{2}\times 4x^{2}-4\times 14400=0$$

Reorder the terms.

$$-\frac{1}{4}\times 4x^{2}x^{2}+4\times 169x^{2}-4\times 14400=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-\frac{1}{4}\times 4x^{4}+4\times 169x^{2}-4\times 14400=0$$

Do the multiplications.

$$-\frac{1}{4}\times 4x^{4}+676x^{2}-57600=0$$

Multiply -\frac{1}{4} and 4 to get -1.

$$-x^{4}+676x^{2}-57600=0$$

Substitute t for x^{2}.

$$-t^{2}+676t-57600=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -1 for a, 676 for b, and -57600 for c in the quadratic formula.

$$t=\frac{-676±\sqrt{676^{2}-4\left(-1\right)\left(-57600\right)}}{-2}$$

Do the calculations.

$$t=\frac{-676±476}{-2}$$

Solve the equation t=\frac{-676±476}{-2} when ± is plus and when ± is minus.

$$t=100 t=576$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=10 x=-10 x=24 x=-24$$