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

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

Solve for x

$$x=-6$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of x+5,x+3.

$$\left(x+3\right)\left(x+1\right)=\left(x+5\right)\left(2x-3\right)$$

Use the distributive property to multiply x+3 by x+1 and combine like terms.

$$x^{2}+4x+3=\left(x+5\right)\left(2x-3\right)$$

Use the distributive property to multiply x+5 by 2x-3 and combine like terms.

$$x^{2}+4x+3=2x^{2}+7x-15$$

Subtract 2x^{2} from both sides.

$$x^{2}+4x+3-2x^{2}=7x-15$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+4x+3=7x-15$$

Subtract 7x from both sides.

$$-x^{2}+4x+3-7x=-15$$

Combine 4x and -7x to get -3x.

$$-x^{2}-3x+3=-15$$

Add 15 to both sides.

$$-x^{2}-3x+3+15=0$$

Add 3 and 15 to get 18.

$$-x^{2}-3x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 18}}{2\left(-1\right)}$$

Multiply 4 times 18.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\left(-1\right)}$$

Add 9 to 72.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-3\right)±9}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{3±9}{-2}$$

Now solve the equation x=\frac{3±9}{-2} when ± is plus. Add 3 to 9.

$$x=\frac{12}{-2}$$

Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{3±9}{-2} when ± is minus. Subtract 9 from 3.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-6 x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+5\right), the least common multiple of x+5,x+3.

$$\left(x+3\right)\left(x+1\right)=\left(x+5\right)\left(2x-3\right)$$

Use the distributive property to multiply x+3 by x+1 and combine like terms.

$$x^{2}+4x+3=\left(x+5\right)\left(2x-3\right)$$

Use the distributive property to multiply x+5 by 2x-3 and combine like terms.

$$x^{2}+4x+3=2x^{2}+7x-15$$

Subtract 2x^{2} from both sides.

$$x^{2}+4x+3-2x^{2}=7x-15$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+4x+3=7x-15$$

Subtract 7x from both sides.

$$-x^{2}+4x+3-7x=-15$$

Combine 4x and -7x to get -3x.

$$-x^{2}-3x+3=-15$$

Subtract 3 from both sides.

$$-x^{2}-3x=-15-3$$

Subtract 3 from -15 to get -18.

$$-x^{2}-3x=-18$$

Divide both sides by -1.

$$\frac{-x^{2}-3x}{-1}=-\frac{18}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{3}{-1}\right)x=-\frac{18}{-1}$$

Divide -3 by -1.

$$x^{2}+3x=-\frac{18}{-1}$$

Divide -18 by -1.

$$x^{2}+3x=18$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=18+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=18+\frac{9}{4}$$

Add 18 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{81}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{9}{2} x+\frac{3}{2}=-\frac{9}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=3 x=-6$$