Solve x+4/5x+9=x+4/4x-5

$$\frac { x + 4 } { 5 x + 9 } = \frac { x + 4 } { 4 x - 5 }$$

Solve for x

$$x=-14$$

$$x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{9}{5},\frac{5}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-5\right)\left(5x+9\right), the least common multiple of 5x+9,4x-5.

$$\left(4x-5\right)\left(x+4\right)=\left(5x+9\right)\left(x+4\right)$$

Use the distributive property to multiply 4x-5 by x+4 and combine like terms.

$$4x^{2}+11x-20=\left(5x+9\right)\left(x+4\right)$$

Use the distributive property to multiply 5x+9 by x+4 and combine like terms.

$$4x^{2}+11x-20=5x^{2}+29x+36$$

Subtract 5x^{2} from both sides.

$$4x^{2}+11x-20-5x^{2}=29x+36$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+11x-20=29x+36$$

Subtract 29x from both sides.

$$-x^{2}+11x-20-29x=36$$

Combine 11x and -29x to get -18x.

$$-x^{2}-18x-20=36$$

Subtract 36 from both sides.

$$-x^{2}-18x-20-36=0$$

Subtract 36 from -20 to get -56.

$$-x^{2}-18x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -18 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-1\right)\left(-56\right)}}{2\left(-1\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-1\right)\left(-56\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-18\right)±\sqrt{324+4\left(-56\right)}}{2\left(-1\right)}$$

Multiply 4 times -56.

$$x=\frac{-\left(-18\right)±\sqrt{324-224}}{2\left(-1\right)}$$

Add 324 to -224.

$$x=\frac{-\left(-18\right)±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-\left(-18\right)±10}{2\left(-1\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{18±10}{-2}$$

Now solve the equation x=\frac{18±10}{-2} when ± is plus. Add 18 to 10.

$$x=\frac{28}{-2}$$

Divide 28 by -2.

$$x=-14$$

Now solve the equation x=\frac{18±10}{-2} when ± is minus. Subtract 10 from 18.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

The equation is now solved.

$$x=-14 x=-4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{9}{5},\frac{5}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-5\right)\left(5x+9\right), the least common multiple of 5x+9,4x-5.

$$\left(4x-5\right)\left(x+4\right)=\left(5x+9\right)\left(x+4\right)$$

Use the distributive property to multiply 4x-5 by x+4 and combine like terms.

$$4x^{2}+11x-20=\left(5x+9\right)\left(x+4\right)$$

Use the distributive property to multiply 5x+9 by x+4 and combine like terms.

$$4x^{2}+11x-20=5x^{2}+29x+36$$

Subtract 5x^{2} from both sides.

$$4x^{2}+11x-20-5x^{2}=29x+36$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+11x-20=29x+36$$

Subtract 29x from both sides.

$$-x^{2}+11x-20-29x=36$$

Combine 11x and -29x to get -18x.

$$-x^{2}-18x-20=36$$

Add 20 to both sides.

$$-x^{2}-18x=36+20$$

Add 36 and 20 to get 56.

$$-x^{2}-18x=56$$

Divide both sides by -1.

$$\frac{-x^{2}-18x}{-1}=\frac{56}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{18}{-1}\right)x=\frac{56}{-1}$$

Divide -18 by -1.

$$x^{2}+18x=\frac{56}{-1}$$

Divide 56 by -1.

$$x^{2}+18x=-56$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=-56+9^{2}$$

Square 9.

$$x^{2}+18x+81=-56+81$$

Add -56 to 81.

$$x^{2}+18x+81=25$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+9=5 x+9=-5$$

Subtract 9 from both sides of the equation.

$$x=-4 x=-14$$