Solve x+3/x+5+2/x+1=13/5

$$\frac { x + 3 } { x + 5 } + \frac { 2 } { x + 1 } = \frac { 13 } { 5 }$$

Solve for x

$$x=-6$$

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-1 since division by zero is not defined. Multiply both sides of the equation by 5\left(x+1\right)\left(x+5\right), the least common multiple of x+5,x+1,5.

$$\left(5x+5\right)\left(x+3\right)+\left(5x+25\right)\times 2=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 5x+5 by x+3 and combine like terms.

$$5x^{2}+20x+15+\left(5x+25\right)\times 2=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 5x+25 by 2.

$$5x^{2}+20x+15+10x+50=13\left(x+1\right)\left(x+5\right)$$

Combine 20x and 10x to get 30x.

$$5x^{2}+30x+15+50=13\left(x+1\right)\left(x+5\right)$$

Add 15 and 50 to get 65.

$$5x^{2}+30x+65=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 13 by x+1.

$$5x^{2}+30x+65=\left(13x+13\right)\left(x+5\right)$$

Use the distributive property to multiply 13x+13 by x+5 and combine like terms.

$$5x^{2}+30x+65=13x^{2}+78x+65$$

Subtract 13x^{2} from both sides.

$$5x^{2}+30x+65-13x^{2}=78x+65$$

Combine 5x^{2} and -13x^{2} to get -8x^{2}.

$$-8x^{2}+30x+65=78x+65$$

Subtract 78x from both sides.

$$-8x^{2}+30x+65-78x=65$$

Combine 30x and -78x to get -48x.

$$-8x^{2}-48x+65=65$$

Subtract 65 from both sides.

$$-8x^{2}-48x+65-65=0$$

Subtract 65 from 65 to get 0.

$$-8x^{2}-48x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -48 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}}}{2\left(-8\right)}$$

Take the square root of \left(-48\right)^{2}.

$$x=\frac{-\left(-48\right)±48}{2\left(-8\right)}$$

The opposite of -48 is 48.

$$x=\frac{48±48}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{48±48}{-16}$$

Now solve the equation x=\frac{48±48}{-16} when ± is plus. Add 48 to 48.

$$x=\frac{96}{-16}$$

Divide 96 by -16.

$$x=-6$$

Now solve the equation x=\frac{48±48}{-16} when ± is minus. Subtract 48 from 48.

$$x=\frac{0}{-16}$$

Divide 0 by -16.

$$x=0$$

The equation is now solved.

$$x=-6 x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-1 since division by zero is not defined. Multiply both sides of the equation by 5\left(x+1\right)\left(x+5\right), the least common multiple of x+5,x+1,5.

$$\left(5x+5\right)\left(x+3\right)+\left(5x+25\right)\times 2=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 5x+5 by x+3 and combine like terms.

$$5x^{2}+20x+15+\left(5x+25\right)\times 2=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 5x+25 by 2.

$$5x^{2}+20x+15+10x+50=13\left(x+1\right)\left(x+5\right)$$

Combine 20x and 10x to get 30x.

$$5x^{2}+30x+15+50=13\left(x+1\right)\left(x+5\right)$$

Add 15 and 50 to get 65.

$$5x^{2}+30x+65=13\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 13 by x+1.

$$5x^{2}+30x+65=\left(13x+13\right)\left(x+5\right)$$

Use the distributive property to multiply 13x+13 by x+5 and combine like terms.

$$5x^{2}+30x+65=13x^{2}+78x+65$$

Subtract 13x^{2} from both sides.

$$5x^{2}+30x+65-13x^{2}=78x+65$$

Combine 5x^{2} and -13x^{2} to get -8x^{2}.

$$-8x^{2}+30x+65=78x+65$$

Subtract 78x from both sides.

$$-8x^{2}+30x+65-78x=65$$

Combine 30x and -78x to get -48x.

$$-8x^{2}-48x+65=65$$

Subtract 65 from both sides.

$$-8x^{2}-48x=65-65$$

Subtract 65 from 65 to get 0.

$$-8x^{2}-48x=0$$

Divide both sides by -8.

$$\frac{-8x^{2}-48x}{-8}=\frac{0}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{48}{-8}\right)x=\frac{0}{-8}$$

Divide -48 by -8.

$$x^{2}+6x=\frac{0}{-8}$$

Divide 0 by -8.

$$x^{2}+6x=0$$

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

$$x^{2}+6x+3^{2}=3^{2}$$

Square 3.

$$x^{2}+6x+9=9$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+3=3 x+3=-3$$

Subtract 3 from both sides of the equation.

$$x=0 x=-6$$