Solve 1/x+1/x+3=3

$$\frac{ 1 }{ x } + \frac{ 1 }{ x+3 } =3$$

Solve for x

$$x=\frac{\sqrt{85}-7}{6}\approx 0.369924076$$

$$x=\frac{-\sqrt{85}-7}{6}\approx -2.70325741$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x,x+3.

$$x+3+x=3x\left(x+3\right)$$

Combine x and x to get 2x.

$$2x+3=3x\left(x+3\right)$$

Use the distributive property to multiply 3x by x+3.

$$2x+3=3x^{2}+9x$$

Subtract 3x^{2} from both sides.

$$2x+3-3x^{2}=9x$$

Subtract 9x from both sides.

$$2x+3-3x^{2}-9x=0$$

Combine 2x and -9x to get -7x.

$$-7x+3-3x^{2}=0$$

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.

$$-3x^{2}-7x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -7 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 3}}{2\left(-3\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 3}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 3}}{2\left(-3\right)}$$

Multiply 12 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49+36}}{2\left(-3\right)}$$

Add 49 to 36.

$$x=\frac{-\left(-7\right)±\sqrt{85}}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{85}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±\sqrt{85}}{-6}$$

Now solve the equation x=\frac{7±\sqrt{85}}{-6} when ± is plus. Add 7 to \sqrt{85}.

$$x=\frac{\sqrt{85}+7}{-6}$$

Divide 7+\sqrt{85} by -6.

$$x=\frac{-\sqrt{85}-7}{6}$$

Now solve the equation x=\frac{7±\sqrt{85}}{-6} when ± is minus. Subtract \sqrt{85} from 7.

$$x=\frac{7-\sqrt{85}}{-6}$$

Divide 7-\sqrt{85} by -6.

$$x=\frac{\sqrt{85}-7}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{85}-7}{6} x=\frac{\sqrt{85}-7}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x,x+3.

$$x+3+x=3x\left(x+3\right)$$

Combine x and x to get 2x.

$$2x+3=3x\left(x+3\right)$$

Use the distributive property to multiply 3x by x+3.

$$2x+3=3x^{2}+9x$$

Subtract 3x^{2} from both sides.

$$2x+3-3x^{2}=9x$$

Subtract 9x from both sides.

$$2x+3-3x^{2}-9x=0$$

Combine 2x and -9x to get -7x.

$$-7x+3-3x^{2}=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$-7x-3x^{2}=-3$$

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.

$$-3x^{2}-7x=-3$$

Divide both sides by -3.

$$\frac{-3x^{2}-7x}{-3}=-\frac{3}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{3}{-3}$$

Divide -7 by -3.

$$x^{2}+\frac{7}{3}x=-\frac{3}{-3}$$

Divide -3 by -3.

$$x^{2}+\frac{7}{3}x=1$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=1+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=1+\frac{49}{36}$$

Add 1 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{85}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{85}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{85}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{\sqrt{85}}{6} x+\frac{7}{6}=-\frac{\sqrt{85}}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$x=\frac{\sqrt{85}-7}{6} x=\frac{-\sqrt{85}-7}{6}$$