Solve x^2/left(10-xright)^2+64=9/64

$$\frac{ { x }^{ 2 } }{ { \left(10-x \right) }^{ 2 } +64 } = \frac{ 9 }{ 64 }$$

Solve for x

$$x=\frac{24\sqrt{155}}{55}-\frac{18}{11}\approx 3.796319825$$

$$x=-\frac{24\sqrt{155}}{55}-\frac{18}{11}\approx -7.069047097$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 64\left(x^{2}-20x+164\right), the least common multiple of \left(10-x\right)^{2}+64,64.

$$64x^{2}=9\left(x^{2}-20x+164\right)$$

Use the distributive property to multiply 9 by x^{2}-20x+164.

$$64x^{2}=9x^{2}-180x+1476$$

Subtract 9x^{2} from both sides.

$$64x^{2}-9x^{2}=-180x+1476$$

Combine 64x^{2} and -9x^{2} to get 55x^{2}.

$$55x^{2}=-180x+1476$$

Add 180x to both sides.

$$55x^{2}+180x=1476$$

Subtract 1476 from both sides.

$$55x^{2}+180x-1476=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 55 for a, 180 for b, and -1476 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-180±\sqrt{180^{2}-4\times 55\left(-1476\right)}}{2\times 55}$$

Square 180.

$$x=\frac{-180±\sqrt{32400-4\times 55\left(-1476\right)}}{2\times 55}$$

Multiply -4 times 55.

$$x=\frac{-180±\sqrt{32400-220\left(-1476\right)}}{2\times 55}$$

Multiply -220 times -1476.

$$x=\frac{-180±\sqrt{32400+324720}}{2\times 55}$$

Add 32400 to 324720.

$$x=\frac{-180±\sqrt{357120}}{2\times 55}$$

Take the square root of 357120.

$$x=\frac{-180±48\sqrt{155}}{2\times 55}$$

Multiply 2 times 55.

$$x=\frac{-180±48\sqrt{155}}{110}$$

Now solve the equation x=\frac{-180±48\sqrt{155}}{110} when ± is plus. Add -180 to 48\sqrt{155}.

$$x=\frac{48\sqrt{155}-180}{110}$$

Divide -180+48\sqrt{155} by 110.

$$x=\frac{24\sqrt{155}}{55}-\frac{18}{11}$$

Now solve the equation x=\frac{-180±48\sqrt{155}}{110} when ± is minus. Subtract 48\sqrt{155} from -180.

$$x=\frac{-48\sqrt{155}-180}{110}$$

Divide -180-48\sqrt{155} by 110.

$$x=-\frac{24\sqrt{155}}{55}-\frac{18}{11}$$

The equation is now solved.

$$x=\frac{24\sqrt{155}}{55}-\frac{18}{11} x=-\frac{24\sqrt{155}}{55}-\frac{18}{11}$$

Steps for Completing the Square

Multiply both sides of the equation by 64\left(x^{2}-20x+164\right), the least common multiple of \left(10-x\right)^{2}+64,64.

$$64x^{2}=9\left(x^{2}-20x+164\right)$$

Use the distributive property to multiply 9 by x^{2}-20x+164.

$$64x^{2}=9x^{2}-180x+1476$$

Subtract 9x^{2} from both sides.

$$64x^{2}-9x^{2}=-180x+1476$$

Combine 64x^{2} and -9x^{2} to get 55x^{2}.

$$55x^{2}=-180x+1476$$

Add 180x to both sides.

$$55x^{2}+180x=1476$$

Divide both sides by 55.

$$\frac{55x^{2}+180x}{55}=\frac{1476}{55}$$

Dividing by 55 undoes the multiplication by 55.

$$x^{2}+\frac{180}{55}x=\frac{1476}{55}$$

Reduce the fraction \frac{180}{55} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{36}{11}x=\frac{1476}{55}$$

Divide \frac{36}{11}, the coefficient of the x term, by 2 to get \frac{18}{11}. Then add the square of \frac{18}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{36}{11}x+\left(\frac{18}{11}\right)^{2}=\frac{1476}{55}+\left(\frac{18}{11}\right)^{2}$$

Square \frac{18}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{36}{11}x+\frac{324}{121}=\frac{1476}{55}+\frac{324}{121}$$

Add \frac{1476}{55} to \frac{324}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{36}{11}x+\frac{324}{121}=\frac{17856}{605}$$

Factor x^{2}+\frac{36}{11}x+\frac{324}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{18}{11}\right)^{2}=\frac{17856}{605}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{18}{11}\right)^{2}}=\sqrt{\frac{17856}{605}}$$

Simplify.

$$x+\frac{18}{11}=\frac{24\sqrt{155}}{55} x+\frac{18}{11}=-\frac{24\sqrt{155}}{55}$$

Subtract \frac{18}{11} from both sides of the equation.

$$x=\frac{24\sqrt{155}}{55}-\frac{18}{11} x=-\frac{24\sqrt{155}}{55}-\frac{18}{11}$$