Solve 5/x+4=x/3(x+1)

$$\frac { 5 } { x + 4 } = \frac { x } { 3 ( x + 1 ) }$$

Solve for x

$$x = \frac{\sqrt{181} + 11}{2} \approx 12.226812024$$

$$x=\frac{11-\sqrt{181}}{2}\approx -1.226812024$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+4\right), the least common multiple of x+4,3\left(x+1\right).

$$\left(3x+3\right)\times 5=\left(x+4\right)x$$

Use the distributive property to multiply 3x+3 by 5.

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

Use the distributive property to multiply x+4 by x.

$$15x+15=x^{2}+4x$$

Subtract x^{2} from both sides.

$$15x+15-x^{2}=4x$$

Subtract 4x from both sides.

$$15x+15-x^{2}-4x=0$$

Combine 15x and -4x to get 11x.

$$11x+15-x^{2}=0$$

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.

$$-x^{2}+11x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$x=\frac{-11±\sqrt{121+60}}{2\left(-1\right)}$$

Add 121 to 60.

$$x=\frac{-11±\sqrt{181}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±\sqrt{181}}{-2}$$

Now solve the equation x=\frac{-11±\sqrt{181}}{-2} when ± is plus. Add -11 to \sqrt{181}.

$$x=\frac{\sqrt{181}-11}{-2}$$

Divide -11+\sqrt{181} by -2.

$$x=\frac{11-\sqrt{181}}{2}$$

Now solve the equation x=\frac{-11±\sqrt{181}}{-2} when ± is minus. Subtract \sqrt{181} from -11.

$$x=\frac{-\sqrt{181}-11}{-2}$$

Divide -11-\sqrt{181} by -2.

$$x=\frac{\sqrt{181}+11}{2}$$

The equation is now solved.

$$x=\frac{11-\sqrt{181}}{2} x=\frac{\sqrt{181}+11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+4\right), the least common multiple of x+4,3\left(x+1\right).

$$\left(3x+3\right)\times 5=\left(x+4\right)x$$

Use the distributive property to multiply 3x+3 by 5.

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

Use the distributive property to multiply x+4 by x.

$$15x+15=x^{2}+4x$$

Subtract x^{2} from both sides.

$$15x+15-x^{2}=4x$$

Subtract 4x from both sides.

$$15x+15-x^{2}-4x=0$$

Combine 15x and -4x to get 11x.

$$11x+15-x^{2}=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$11x-x^{2}=-15$$

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.

$$-x^{2}+11x=-15$$

Divide both sides by -1.

$$\frac{-x^{2}+11x}{-1}=-\frac{15}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{11}{-1}x=-\frac{15}{-1}$$

Divide 11 by -1.

$$x^{2}-11x=-\frac{15}{-1}$$

Divide -15 by -1.

$$x^{2}-11x=15$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=15+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=15+\frac{121}{4}$$

Add 15 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{181}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{181}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{181}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{\sqrt{181}}{2} x-\frac{11}{2}=-\frac{\sqrt{181}}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{\sqrt{181}+11}{2} x=\frac{11-\sqrt{181}}{2}$$