Solve 4x+5/5-3x+2/4=1/x-9

$$\frac { 4 x + 5 } { 5 } - \frac { 3 x + 2 } { 4 } = \frac { 1 } { x - 9 }$$

Solve for x

$$x=-11$$

$$x=10$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 9 since division by zero is not defined. Multiply both sides of the equation by 20\left(x-9\right), the least common multiple of 5,4,x-9.

$$\left(4x-36\right)\left(4x+5\right)-\left(5x-45\right)\left(3x+2\right)=20$$

Use the distributive property to multiply 4x-36 by 4x+5 and combine like terms.

$$16x^{2}-124x-180-\left(5x-45\right)\left(3x+2\right)=20$$

Use the distributive property to multiply 5x-45 by 3x+2 and combine like terms.

$$16x^{2}-124x-180-\left(15x^{2}-125x-90\right)=20$$

To find the opposite of 15x^{2}-125x-90, find the opposite of each term.

$$16x^{2}-124x-180-15x^{2}+125x+90=20$$

Combine 16x^{2} and -15x^{2} to get x^{2}.

$$x^{2}-124x-180+125x+90=20$$

Combine -124x and 125x to get x.

$$x^{2}+x-180+90=20$$

Add -180 and 90 to get -90.

$$x^{2}+x-90=20$$

Subtract 20 from both sides.

$$x^{2}+x-90-20=0$$

Subtract 20 from -90 to get -110.

$$x^{2}+x-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-110\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-110\right)}}{2}$$

Multiply -4 times -110.

$$x=\frac{-1±\sqrt{1+440}}{2}$$

Add 1 to 440.

$$x=\frac{-1±\sqrt{441}}{2}$$

Take the square root of 441.

$$x=\frac{-1±21}{2}$$

Now solve the equation x=\frac{-1±21}{2} when ± is plus. Add -1 to 21.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{-1±21}{2} when ± is minus. Subtract 21 from -1.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=10 x=-11$$

Steps for Completing the Square

Variable x cannot be equal to 9 since division by zero is not defined. Multiply both sides of the equation by 20\left(x-9\right), the least common multiple of 5,4,x-9.

$$\left(4x-36\right)\left(4x+5\right)-\left(5x-45\right)\left(3x+2\right)=20$$

Use the distributive property to multiply 4x-36 by 4x+5 and combine like terms.

$$16x^{2}-124x-180-\left(5x-45\right)\left(3x+2\right)=20$$

Use the distributive property to multiply 5x-45 by 3x+2 and combine like terms.

$$16x^{2}-124x-180-\left(15x^{2}-125x-90\right)=20$$

To find the opposite of 15x^{2}-125x-90, find the opposite of each term.

$$16x^{2}-124x-180-15x^{2}+125x+90=20$$

Combine 16x^{2} and -15x^{2} to get x^{2}.

$$x^{2}-124x-180+125x+90=20$$

Combine -124x and 125x to get x.

$$x^{2}+x-180+90=20$$

Add -180 and 90 to get -90.

$$x^{2}+x-90=20$$

Add 90 to both sides.

$$x^{2}+x=20+90$$

Add 20 and 90 to get 110.

$$x^{2}+x=110$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=110+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=110+\frac{1}{4}$$

Add 110 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{441}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{441}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{441}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{21}{2} x+\frac{1}{2}=-\frac{21}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=10 x=-11$$