Solve 4x^-2(25-x)=85

$$4 x ^ { - 2 } ( 25 - x ) = 85$$

Solve for x

$$x = \frac{2 \sqrt{2126} - 2}{85} \approx 1.06137806$$

$$x=\frac{-2\sqrt{2126}-2}{85}\approx -1.108436883$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x^{-2} by 25-x.

$$100x^{-2}-4x^{-1}=85$$

Subtract 85 from both sides.

$$100x^{-2}-4x^{-1}-85=0$$

Reorder the terms.

$$-85-4\times \frac{1}{x}+100x^{-2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$x\left(-85\right)-4+100x^{-2}x=0$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$x\left(-85\right)-4+100x^{-1}=0$$

Reorder the terms.

$$-85x-4+100\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-85xx+x\left(-4\right)+100\times 1=0$$

Multiply x and x to get x^{2}.

$$-85x^{2}+x\left(-4\right)+100\times 1=0$$

Multiply 100 and 1 to get 100.

$$-85x^{2}+x\left(-4\right)+100=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-85x^{2}-4x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -85 for a, -4 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-85\right)\times 100}}{2\left(-85\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-85\right)\times 100}}{2\left(-85\right)}$$

Multiply -4 times -85.

$$x=\frac{-\left(-4\right)±\sqrt{16+340\times 100}}{2\left(-85\right)}$$

Multiply 340 times 100.

$$x=\frac{-\left(-4\right)±\sqrt{16+34000}}{2\left(-85\right)}$$

Add 16 to 34000.

$$x=\frac{-\left(-4\right)±\sqrt{34016}}{2\left(-85\right)}$$

Take the square root of 34016.

$$x=\frac{-\left(-4\right)±4\sqrt{2126}}{2\left(-85\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{2126}}{2\left(-85\right)}$$

Multiply 2 times -85.

$$x=\frac{4±4\sqrt{2126}}{-170}$$

Now solve the equation x=\frac{4±4\sqrt{2126}}{-170} when ± is plus. Add 4 to 4\sqrt{2126}.

$$x=\frac{4\sqrt{2126}+4}{-170}$$

Divide 4+4\sqrt{2126} by -170.

$$x=\frac{-2\sqrt{2126}-2}{85}$$

Now solve the equation x=\frac{4±4\sqrt{2126}}{-170} when ± is minus. Subtract 4\sqrt{2126} from 4.

$$x=\frac{4-4\sqrt{2126}}{-170}$$

Divide 4-4\sqrt{2126} by -170.

$$x=\frac{2\sqrt{2126}-2}{85}$$

The equation is now solved.

$$x=\frac{-2\sqrt{2126}-2}{85} x=\frac{2\sqrt{2126}-2}{85}$$

Steps for Completing the Square

Use the distributive property to multiply 4x^{-2} by 25-x.

$$100x^{-2}-4x^{-1}=85$$

Reorder the terms.

$$-4\times \frac{1}{x}+100x^{-2}=85$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-4+100x^{-2}x=85x$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$-4+100x^{-1}=85x$$

Subtract 85x from both sides.

$$-4+100x^{-1}-85x=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$100x^{-1}-85x=4$$

Reorder the terms.

$$-85x+100\times \frac{1}{x}=4$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-85xx+100\times 1=4x$$

Multiply x and x to get x^{2}.

$$-85x^{2}+100\times 1=4x$$

Multiply 100 and 1 to get 100.

$$-85x^{2}+100=4x$$

Subtract 4x from both sides.

$$-85x^{2}+100-4x=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$-85x^{2}-4x=-100$$

Divide both sides by -85.

$$\frac{-85x^{2}-4x}{-85}=-\frac{100}{-85}$$

Dividing by -85 undoes the multiplication by -85.

$$x^{2}+\left(-\frac{4}{-85}\right)x=-\frac{100}{-85}$$

Divide -4 by -85.

$$x^{2}+\frac{4}{85}x=-\frac{100}{-85}$$

Reduce the fraction \frac{-100}{-85} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{4}{85}x=\frac{20}{17}$$

Divide \frac{4}{85}, the coefficient of the x term, by 2 to get \frac{2}{85}. Then add the square of \frac{2}{85} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{85}x+\left(\frac{2}{85}\right)^{2}=\frac{20}{17}+\left(\frac{2}{85}\right)^{2}$$

Square \frac{2}{85} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{85}x+\frac{4}{7225}=\frac{20}{17}+\frac{4}{7225}$$

Add \frac{20}{17} to \frac{4}{7225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{85}x+\frac{4}{7225}=\frac{8504}{7225}$$

Factor x^{2}+\frac{4}{85}x+\frac{4}{7225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{85}\right)^{2}=\frac{8504}{7225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{85}\right)^{2}}=\sqrt{\frac{8504}{7225}}$$

Simplify.

$$x+\frac{2}{85}=\frac{2\sqrt{2126}}{85} x+\frac{2}{85}=-\frac{2\sqrt{2126}}{85}$$

Subtract \frac{2}{85} from both sides of the equation.

$$x=\frac{2\sqrt{2126}-2}{85} x=\frac{-2\sqrt{2126}-2}{85}$$