Solve x+8/20x=27/20

$$x + \frac { 8 } { 20 x } = \frac { 27 } { 20 }$$

Solve for x

$$x=\frac{\sqrt{89}+27}{40}\approx 0.910849528$$

$$x=\frac{27-\sqrt{89}}{40}\approx 0.439150472$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 20x, the least common multiple of 20x,20.

$$20xx+8=27x$$

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

$$20x^{2}+8=27x$$

Subtract 27x from both sides.

$$20x^{2}+8-27x=0$$

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.

$$20x^{2}-27x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -27 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 20\times 8}}{2\times 20}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\times 20\times 8}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-27\right)±\sqrt{729-80\times 8}}{2\times 20}$$

Multiply -80 times 8.

$$x=\frac{-\left(-27\right)±\sqrt{729-640}}{2\times 20}$$

Add 729 to -640.

$$x=\frac{-\left(-27\right)±\sqrt{89}}{2\times 20}$$

The opposite of -27 is 27.

$$x=\frac{27±\sqrt{89}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{27±\sqrt{89}}{40}$$

Now solve the equation x=\frac{27±\sqrt{89}}{40} when ± is plus. Add 27 to \sqrt{89}.

$$x=\frac{\sqrt{89}+27}{40}$$

Now solve the equation x=\frac{27±\sqrt{89}}{40} when ± is minus. Subtract \sqrt{89} from 27.

$$x=\frac{27-\sqrt{89}}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{89}+27}{40} x=\frac{27-\sqrt{89}}{40}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 20x, the least common multiple of 20x,20.

$$20xx+8=27x$$

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

$$20x^{2}+8=27x$$

Subtract 27x from both sides.

$$20x^{2}+8-27x=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$20x^{2}-27x=-8$$

Divide both sides by 20.

$$\frac{20x^{2}-27x}{20}=-\frac{8}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}-\frac{27}{20}x=-\frac{8}{20}$$

Reduce the fraction \frac{-8}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{27}{20}x=-\frac{2}{5}$$

Divide -\frac{27}{20}, the coefficient of the x term, by 2 to get -\frac{27}{40}. Then add the square of -\frac{27}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{27}{20}x+\left(-\frac{27}{40}\right)^{2}=-\frac{2}{5}+\left(-\frac{27}{40}\right)^{2}$$

Square -\frac{27}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{27}{20}x+\frac{729}{1600}=-\frac{2}{5}+\frac{729}{1600}$$

Add -\frac{2}{5} to \frac{729}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{27}{20}x+\frac{729}{1600}=\frac{89}{1600}$$

Factor x^{2}-\frac{27}{20}x+\frac{729}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{27}{40}\right)^{2}=\frac{89}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{40}\right)^{2}}=\sqrt{\frac{89}{1600}}$$

Simplify.

$$x-\frac{27}{40}=\frac{\sqrt{89}}{40} x-\frac{27}{40}=-\frac{\sqrt{89}}{40}$$

Add \frac{27}{40} to both sides of the equation.

$$x=\frac{\sqrt{89}+27}{40} x=\frac{27-\sqrt{89}}{40}$$