Solve x-2x+5/x-9=7/x-9

$$x - \frac { 2 x + 5 } { x - 9 } = \frac { 7 } { x - 9 }$$

Solve for x

$$x=-1$$

$$x=12$$

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 x-9.

$$\left(x-9\right)x-\left(2x+5\right)=7$$

Use the distributive property to multiply x-9 by x.

$$x^{2}-9x-\left(2x+5\right)=7$$

To find the opposite of 2x+5, find the opposite of each term.

$$x^{2}-9x-2x-5=7$$

Combine -9x and -2x to get -11x.

$$x^{2}-11x-5=7$$

Subtract 7 from both sides.

$$x^{2}-11x-5-7=0$$

Subtract 7 from -5 to get -12.

$$x^{2}-11x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-12\right)}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-12\right)}}{2}$$

Multiply -4 times -12.

$$x=\frac{-\left(-11\right)±\sqrt{121+48}}{2}$$

Add 121 to 48.

$$x=\frac{-\left(-11\right)±\sqrt{169}}{2}$$

Take the square root of 169.

$$x=\frac{-\left(-11\right)±13}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±13}{2}$$

Now solve the equation x=\frac{11±13}{2} when ± is plus. Add 11 to 13.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{11±13}{2} when ± is minus. Subtract 13 from 11.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=12 x=-1$$

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 x-9.

$$\left(x-9\right)x-\left(2x+5\right)=7$$

Use the distributive property to multiply x-9 by x.

$$x^{2}-9x-\left(2x+5\right)=7$$

To find the opposite of 2x+5, find the opposite of each term.

$$x^{2}-9x-2x-5=7$$

Combine -9x and -2x to get -11x.

$$x^{2}-11x-5=7$$

Add 5 to both sides.

$$x^{2}-11x=7+5$$

Add 7 and 5 to get 12.

$$x^{2}-11x=12$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=12+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=12+\frac{121}{4}$$

Add 12 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{169}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{13}{2} x-\frac{11}{2}=-\frac{13}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=12 x=-1$$