Solve x^2/left(x-16right)^2=3/5

$$\frac{ { x }^{ 2 } }{ { \left(x-16 \right) }^{ 2 } } = \frac{ 3 }{ 5 }$$

Solve for x

$$x=8\sqrt{15}-24\approx 6.98386677$$

$$x=-8\sqrt{15}-24\approx -54.98386677$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 16 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-16\right)^{2}, the least common multiple of \left(x-16\right)^{2},5.

$$5x^{2}=3\left(x-16\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-16\right)^{2}.

$$5x^{2}=3\left(x^{2}-32x+256\right)$$

Use the distributive property to multiply 3 by x^{2}-32x+256.

$$5x^{2}=3x^{2}-96x+768$$

Subtract 3x^{2} from both sides.

$$5x^{2}-3x^{2}=-96x+768$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}=-96x+768$$

Add 96x to both sides.

$$2x^{2}+96x=768$$

Subtract 768 from both sides.

$$2x^{2}+96x-768=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 96 for b, and -768 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-96±\sqrt{96^{2}-4\times 2\left(-768\right)}}{2\times 2}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\times 2\left(-768\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-96±\sqrt{9216-8\left(-768\right)}}{2\times 2}$$

Multiply -8 times -768.

$$x=\frac{-96±\sqrt{9216+6144}}{2\times 2}$$

Add 9216 to 6144.

$$x=\frac{-96±\sqrt{15360}}{2\times 2}$$

Take the square root of 15360.

$$x=\frac{-96±32\sqrt{15}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-96±32\sqrt{15}}{4}$$

Now solve the equation x=\frac{-96±32\sqrt{15}}{4} when ± is plus. Add -96 to 32\sqrt{15}.

$$x=\frac{32\sqrt{15}-96}{4}$$

Divide -96+32\sqrt{15} by 4.

$$x=8\sqrt{15}-24$$

Now solve the equation x=\frac{-96±32\sqrt{15}}{4} when ± is minus. Subtract 32\sqrt{15} from -96.

$$x=\frac{-32\sqrt{15}-96}{4}$$

Divide -96-32\sqrt{15} by 4.

$$x=-8\sqrt{15}-24$$

The equation is now solved.

$$x=8\sqrt{15}-24 x=-8\sqrt{15}-24$$

Steps for Completing the Square

Variable x cannot be equal to 16 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-16\right)^{2}, the least common multiple of \left(x-16\right)^{2},5.

$$5x^{2}=3\left(x-16\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-16\right)^{2}.

$$5x^{2}=3\left(x^{2}-32x+256\right)$$

Use the distributive property to multiply 3 by x^{2}-32x+256.

$$5x^{2}=3x^{2}-96x+768$$

Subtract 3x^{2} from both sides.

$$5x^{2}-3x^{2}=-96x+768$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}=-96x+768$$

Add 96x to both sides.

$$2x^{2}+96x=768$$

Divide both sides by 2.

$$\frac{2x^{2}+96x}{2}=\frac{768}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{96}{2}x=\frac{768}{2}$$

Divide 96 by 2.

$$x^{2}+48x=\frac{768}{2}$$

Divide 768 by 2.

$$x^{2}+48x=384$$

Divide 48, the coefficient of the x term, by 2 to get 24. Then add the square of 24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+48x+24^{2}=384+24^{2}$$

Square 24.

$$x^{2}+48x+576=384+576$$

Add 384 to 576.

$$x^{2}+48x+576=960$$

Factor x^{2}+48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+24\right)^{2}=960$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{960}$$

Simplify.

$$x+24=8\sqrt{15} x+24=-8\sqrt{15}$$

Subtract 24 from both sides of the equation.

$$x=8\sqrt{15}-24 x=-8\sqrt{15}-24$$