Solve x+5/x+6=3x-13/2x-3

$$\frac{ x+5 }{ x+6 } = \frac{ 3x-13 }{ 2x-3 }$$

Solve for x

$$x=-7$$

$$x=9$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-3\right)\left(x+6\right), the least common multiple of x+6,2x-3.

$$\left(2x-3\right)\left(x+5\right)=\left(x+6\right)\left(3x-13\right)$$

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

$$2x^{2}+7x-15=\left(x+6\right)\left(3x-13\right)$$

Use the distributive property to multiply x+6 by 3x-13 and combine like terms.

$$2x^{2}+7x-15=3x^{2}+5x-78$$

Subtract 3x^{2} from both sides.

$$2x^{2}+7x-15-3x^{2}=5x-78$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}+7x-15=5x-78$$

Subtract 5x from both sides.

$$-x^{2}+7x-15-5x=-78$$

Combine 7x and -5x to get 2x.

$$-x^{2}+2x-15=-78$$

Add 78 to both sides.

$$-x^{2}+2x-15+78=0$$

Add -15 and 78 to get 63.

$$-x^{2}+2x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 63}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 63}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 63}}{2\left(-1\right)}$$

Multiply 4 times 63.

$$x=\frac{-2±\sqrt{4+252}}{2\left(-1\right)}$$

Add 4 to 252.

$$x=\frac{-2±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$x=\frac{-2±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±16}{-2}$$

Now solve the equation x=\frac{-2±16}{-2} when ± is plus. Add -2 to 16.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{-2±16}{-2} when ± is minus. Subtract 16 from -2.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=-7 x=9$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-3\right)\left(x+6\right), the least common multiple of x+6,2x-3.

$$\left(2x-3\right)\left(x+5\right)=\left(x+6\right)\left(3x-13\right)$$

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

$$2x^{2}+7x-15=\left(x+6\right)\left(3x-13\right)$$

Use the distributive property to multiply x+6 by 3x-13 and combine like terms.

$$2x^{2}+7x-15=3x^{2}+5x-78$$

Subtract 3x^{2} from both sides.

$$2x^{2}+7x-15-3x^{2}=5x-78$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}+7x-15=5x-78$$

Subtract 5x from both sides.

$$-x^{2}+7x-15-5x=-78$$

Combine 7x and -5x to get 2x.

$$-x^{2}+2x-15=-78$$

Add 15 to both sides.

$$-x^{2}+2x=-78+15$$

Add -78 and 15 to get -63.

$$-x^{2}+2x=-63$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{63}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{63}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{63}{-1}$$

Divide -63 by -1.

$$x^{2}-2x=63$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=63+1$$

Add 63 to 1.

$$x^{2}-2x+1=64$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-1=8 x-1=-8$$

Add 1 to both sides of the equation.

$$x=9 x=-7$$