Solve 22/x^2-5.8/left(2-xright)^2=0

$$\frac{ 22 }{ { x }^{ 2 } } - \frac{ 5.8 }{ { \left(2-x \right) }^{ 2 } } =0$$

Solve for x

$$x = \frac{2 \sqrt{3190} + 220}{81} \approx 4.110619382$$

$$x = \frac{220 - 2 \sqrt{3190}}{81} \approx 1.321479383$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x^{2}\left(x-2\right)^{2}, the least common multiple of x^{2},\left(2-x\right)^{2}.

$$\left(x-2\right)^{2}\times 22-x^{2}\times 5.8=0$$

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

$$\left(x^{2}-4x+4\right)\times 22-x^{2}\times 5.8=0$$

Use the distributive property to multiply x^{2}-4x+4 by 22.

$$22x^{2}-88x+88-x^{2}\times 5.8=0$$

Multiply -1 and 5.8 to get -5.8.

$$22x^{2}-88x+88-5.8x^{2}=0$$

Combine 22x^{2} and -5.8x^{2} to get 16.2x^{2}.

$$16.2x^{2}-88x+88=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16.2 for a, -88 for b, and 88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-88\right)±\sqrt{\left(-88\right)^{2}-4\times 16.2\times 88}}{2\times 16.2}$$

Square -88.

$$x=\frac{-\left(-88\right)±\sqrt{7744-4\times 16.2\times 88}}{2\times 16.2}$$

Multiply -4 times 16.2.

$$x=\frac{-\left(-88\right)±\sqrt{7744-64.8\times 88}}{2\times 16.2}$$

Multiply -64.8 times 88.

$$x=\frac{-\left(-88\right)±\sqrt{7744-5702.4}}{2\times 16.2}$$

Add 7744 to -5702.4.

$$x=\frac{-\left(-88\right)±\sqrt{2041.6}}{2\times 16.2}$$

Take the square root of 2041.6.

$$x=\frac{-\left(-88\right)±\frac{4\sqrt{3190}}{5}}{2\times 16.2}$$

The opposite of -88 is 88.

$$x=\frac{88±\frac{4\sqrt{3190}}{5}}{2\times 16.2}$$

Multiply 2 times 16.2.

$$x=\frac{88±\frac{4\sqrt{3190}}{5}}{32.4}$$

Now solve the equation x=\frac{88±\frac{4\sqrt{3190}}{5}}{32.4} when ± is plus. Add 88 to \frac{4\sqrt{3190}}{5}.

$$x=\frac{\frac{4\sqrt{3190}}{5}+88}{32.4}$$

Divide 88+\frac{4\sqrt{3190}}{5} by 32.4 by multiplying 88+\frac{4\sqrt{3190}}{5} by the reciprocal of 32.4.

$$x=\frac{2\sqrt{3190}+220}{81}$$

Now solve the equation x=\frac{88±\frac{4\sqrt{3190}}{5}}{32.4} when ± is minus. Subtract \frac{4\sqrt{3190}}{5} from 88.

$$x=\frac{-\frac{4\sqrt{3190}}{5}+88}{32.4}$$

Divide 88-\frac{4\sqrt{3190}}{5} by 32.4 by multiplying 88-\frac{4\sqrt{3190}}{5} by the reciprocal of 32.4.

$$x=\frac{220-2\sqrt{3190}}{81}$$

The equation is now solved.

$$x=\frac{2\sqrt{3190}+220}{81} x=\frac{220-2\sqrt{3190}}{81}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x^{2}\left(x-2\right)^{2}, the least common multiple of x^{2},\left(2-x\right)^{2}.

$$\left(x-2\right)^{2}\times 22-x^{2}\times 5.8=0$$

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

$$\left(x^{2}-4x+4\right)\times 22-x^{2}\times 5.8=0$$

Use the distributive property to multiply x^{2}-4x+4 by 22.

$$22x^{2}-88x+88-x^{2}\times 5.8=0$$

Subtract 88 from both sides. Anything subtracted from zero gives its negation.

$$22x^{2}-88x-x^{2}\times 5.8=-88$$

Multiply -1 and 5.8 to get -5.8.

$$22x^{2}-88x-5.8x^{2}=-88$$

Combine 22x^{2} and -5.8x^{2} to get 16.2x^{2}.

$$16.2x^{2}-88x=-88$$

Divide both sides of the equation by 16.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{16.2x^{2}-88x}{16.2}=-\frac{88}{16.2}$$

Dividing by 16.2 undoes the multiplication by 16.2.

$$x^{2}+\left(-\frac{88}{16.2}\right)x=-\frac{88}{16.2}$$

Divide -88 by 16.2 by multiplying -88 by the reciprocal of 16.2.

$$x^{2}-\frac{440}{81}x=-\frac{88}{16.2}$$

Divide -88 by 16.2 by multiplying -88 by the reciprocal of 16.2.

$$x^{2}-\frac{440}{81}x=-\frac{440}{81}$$

Divide -\frac{440}{81}, the coefficient of the x term, by 2 to get -\frac{220}{81}. Then add the square of -\frac{220}{81} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{440}{81}x+\left(-\frac{220}{81}\right)^{2}=-\frac{440}{81}+\left(-\frac{220}{81}\right)^{2}$$

Square -\frac{220}{81} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{440}{81}x+\frac{48400}{6561}=-\frac{440}{81}+\frac{48400}{6561}$$

Add -\frac{440}{81} to \frac{48400}{6561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{440}{81}x+\frac{48400}{6561}=\frac{12760}{6561}$$

Factor x^{2}-\frac{440}{81}x+\frac{48400}{6561}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{220}{81}\right)^{2}=\frac{12760}{6561}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{220}{81}\right)^{2}}=\sqrt{\frac{12760}{6561}}$$

Simplify.

$$x-\frac{220}{81}=\frac{2\sqrt{3190}}{81} x-\frac{220}{81}=-\frac{2\sqrt{3190}}{81}$$

Add \frac{220}{81} to both sides of the equation.

$$x=\frac{2\sqrt{3190}+220}{81} x=\frac{220-2\sqrt{3190}}{81}$$