Solve 2x/x+9=x-2/x-1

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

Solve for x

$$x=3$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+9\right), the least common multiple of x+9,x-1.

$$\left(x-1\right)\times 2x=\left(x+9\right)\left(x-2\right)$$

Use the distributive property to multiply x-1 by 2.

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

Use the distributive property to multiply 2x-2 by x.

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

Use the distributive property to multiply x+9 by x-2 and combine like terms.

$$2x^{2}-2x=x^{2}+7x-18$$

Subtract x^{2} from both sides.

$$2x^{2}-2x-x^{2}=7x-18$$

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

$$x^{2}-2x=7x-18$$

Subtract 7x from both sides.

$$x^{2}-2x-7x=-18$$

Combine -2x and -7x to get -9x.

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

Add 18 to both sides.

$$x^{2}-9x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-9\right)±\sqrt{81-72}}{2}$$

Add 81 to -72.

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

Take the square root of 9.

$$x=\frac{-\left(-9\right)±3}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±3}{2}$$

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

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

Divide 12 by 2.

$$x=6$$

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

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

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 -9,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+9\right), the least common multiple of x+9,x-1.

$$\left(x-1\right)\times 2x=\left(x+9\right)\left(x-2\right)$$

Use the distributive property to multiply x-1 by 2.

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

Use the distributive property to multiply 2x-2 by x.

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

Use the distributive property to multiply x+9 by x-2 and combine like terms.

$$2x^{2}-2x=x^{2}+7x-18$$

Subtract x^{2} from both sides.

$$2x^{2}-2x-x^{2}=7x-18$$

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

$$x^{2}-2x=7x-18$$

Subtract 7x from both sides.

$$x^{2}-2x-7x=-18$$

Combine -2x and -7x to get -9x.

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

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-18+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-18+\frac{81}{4}$$

Add -18 to \frac{81}{4}.

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

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Add \frac{9}{2} to both sides of the equation.

$$x=6 x=3$$