Solve 20/18=x+5/18x

$$\frac { 20 } { 18 } = x + \frac { 5 } { 18 x }$$

Solve for x

$$x=\frac{\sqrt{10}}{18}+\frac{5}{9}\approx 0.731237648$$

$$x=-\frac{\sqrt{10}}{18}+\frac{5}{9}\approx 0.379873463$$

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 18x, the least common multiple of 18,18x.

$$x\times 20=18xx+5$$

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

$$x\times 20=18x^{2}+5$$

Subtract 18x^{2} from both sides.

$$x\times 20-18x^{2}=5$$

Subtract 5 from both sides.

$$x\times 20-18x^{2}-5=0$$

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.

$$-18x^{2}+20x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 20 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-18\right)\left(-5\right)}}{2\left(-18\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-18\right)\left(-5\right)}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-20±\sqrt{400+72\left(-5\right)}}{2\left(-18\right)}$$

Multiply 72 times -5.

$$x=\frac{-20±\sqrt{400-360}}{2\left(-18\right)}$$

Add 400 to -360.

$$x=\frac{-20±\sqrt{40}}{2\left(-18\right)}$$

Take the square root of 40.

$$x=\frac{-20±2\sqrt{10}}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-20±2\sqrt{10}}{-36}$$

Now solve the equation x=\frac{-20±2\sqrt{10}}{-36} when ± is plus. Add -20 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-20}{-36}$$

Divide -20+2\sqrt{10} by -36.

$$x=-\frac{\sqrt{10}}{18}+\frac{5}{9}$$

Now solve the equation x=\frac{-20±2\sqrt{10}}{-36} when ± is minus. Subtract 2\sqrt{10} from -20.

$$x=\frac{-2\sqrt{10}-20}{-36}$$

Divide -20-2\sqrt{10} by -36.

$$x=\frac{\sqrt{10}}{18}+\frac{5}{9}$$

The equation is now solved.

$$x=-\frac{\sqrt{10}}{18}+\frac{5}{9} x=\frac{\sqrt{10}}{18}+\frac{5}{9}$$

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 18x, the least common multiple of 18,18x.

$$x\times 20=18xx+5$$

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

$$x\times 20=18x^{2}+5$$

Subtract 18x^{2} from both sides.

$$x\times 20-18x^{2}=5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-18x^{2}+20x=5$$

Divide both sides by -18.

$$\frac{-18x^{2}+20x}{-18}=\frac{5}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{20}{-18}x=\frac{5}{-18}$$

Reduce the fraction \frac{20}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{10}{9}x=\frac{5}{-18}$$

Divide 5 by -18.

$$x^{2}-\frac{10}{9}x=-\frac{5}{18}$$

Divide -\frac{10}{9}, the coefficient of the x term, by 2 to get -\frac{5}{9}. Then add the square of -\frac{5}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{9}x+\left(-\frac{5}{9}\right)^{2}=-\frac{5}{18}+\left(-\frac{5}{9}\right)^{2}$$

Square -\frac{5}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{9}x+\frac{25}{81}=-\frac{5}{18}+\frac{25}{81}$$

Add -\frac{5}{18} to \frac{25}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{10}{9}x+\frac{25}{81}=\frac{5}{162}$$

Factor x^{2}-\frac{10}{9}x+\frac{25}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{9}\right)^{2}=\frac{5}{162}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{9}\right)^{2}}=\sqrt{\frac{5}{162}}$$

Simplify.

$$x-\frac{5}{9}=\frac{\sqrt{10}}{18} x-\frac{5}{9}=-\frac{\sqrt{10}}{18}$$

Add \frac{5}{9} to both sides of the equation.

$$x=\frac{\sqrt{10}}{18}+\frac{5}{9} x=-\frac{\sqrt{10}}{18}+\frac{5}{9}$$