Solve 1/x+4/x+1+1=15/x

$$\frac { 1 } { x } + \frac { 4 } { x + 1 } + 1 = \frac { 15 } { x }$$

Solve for x

$$x = \frac{\sqrt{137} + 9}{2} \approx 10.352349955$$

$$x=\frac{9-\sqrt{137}}{2}\approx -1.352349955$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x,x+1.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

Combine x and x\times 4 to get 5x.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

Use the distributive property to multiply x by x+1.

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

Combine 5x and x to get 6x.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

Use the distributive property to multiply x+1 by 15.

$$6x+1+x^{2}=15x+15$$

Subtract 15x from both sides.

$$6x+1+x^{2}-15x=15$$

Combine 6x and -15x to get -9x.

$$-9x+1+x^{2}=15$$

Subtract 15 from both sides.

$$-9x+1+x^{2}-15=0$$

Subtract 15 from 1 to get -14.

$$-9x-14+x^{2}=0$$

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.

$$x^{2}-9x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-14\right)}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$x=\frac{-\left(-9\right)±\sqrt{81+56}}{2}$$

Add 81 to 56.

$$x=\frac{-\left(-9\right)±\sqrt{137}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{137}}{2}$$

Now solve the equation x=\frac{9±\sqrt{137}}{2} when ± is plus. Add 9 to \sqrt{137}.

$$x=\frac{\sqrt{137}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{137}}{2} when ± is minus. Subtract \sqrt{137} from 9.

$$x=\frac{9-\sqrt{137}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x,x+1.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

Combine x and x\times 4 to get 5x.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

Use the distributive property to multiply x by x+1.

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

Combine 5x and x to get 6x.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

Use the distributive property to multiply x+1 by 15.

$$6x+1+x^{2}=15x+15$$

Subtract 15x from both sides.

$$6x+1+x^{2}-15x=15$$

Combine 6x and -15x to get -9x.

$$-9x+1+x^{2}=15$$

Subtract 1 from both sides.

$$-9x+x^{2}=15-1$$

Subtract 1 from 15 to get 14.

$$-9x+x^{2}=14$$

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.

$$x^{2}-9x=14$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=14+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=14+\frac{81}{4}$$

Add 14 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{137}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{137}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{137}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{137}}{2} x-\frac{9}{2}=-\frac{\sqrt{137}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$