Solve x+2/x+3-x^2/x^2-9=1-x-1/3-x

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

Solve for x

$$x = \frac{\sqrt{57} - 3}{4} \approx 1.137458609$$

$$x=\frac{-\sqrt{57}-3}{4}\approx -2.637458609$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x+3,x^{2}-9,3-x.

$$\left(x-3\right)\left(x+2\right)-x^{2}=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)\left(x-1\right)\right)$$

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

$$x^{2}-x-6-x^{2}=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)\left(x-1\right)\right)$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-\left(3+x\right)\left(x-1\right)\right)$$

Use the distributive property to multiply -1 by 3+x.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-3-x\right)\left(x-1\right)$$

Use the distributive property to multiply -3-x by x-1 and combine like terms.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-2x+3-x^{2}\right)$$

To find the opposite of -2x+3-x^{2}, find the opposite of each term.

$$x^{2}-x-6-x^{2}=x^{2}-9+2x-3+x^{2}$$

Subtract 3 from -9 to get -12.

$$x^{2}-x-6-x^{2}=x^{2}-12+2x+x^{2}$$

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

$$x^{2}-x-6-x^{2}=2x^{2}-12+2x$$

Subtract 2x^{2} from both sides.

$$x^{2}-x-6-x^{2}-2x^{2}=-12+2x$$

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

$$-x^{2}-x-6-x^{2}=-12+2x$$

Subtract -12 from both sides.

$$-x^{2}-x-6-x^{2}-\left(-12\right)=2x$$

The opposite of -12 is 12.

$$-x^{2}-x-6-x^{2}+12=2x$$

Subtract 2x from both sides.

$$-x^{2}-x-6-x^{2}+12-2x=0$$

Add -6 and 12 to get 6.

$$-x^{2}-x+6-x^{2}-2x=0$$

Combine -x and -2x to get -3x.

$$-x^{2}-3x+6-x^{2}=0$$

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

$$-2x^{2}-3x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -3 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-3\right)±\sqrt{9+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$x=\frac{-\left(-3\right)±\sqrt{9+48}}{2\left(-2\right)}$$

Add 9 to 48.

$$x=\frac{-\left(-3\right)±\sqrt{57}}{2\left(-2\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{57}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{3±\sqrt{57}}{-4}$$

Now solve the equation x=\frac{3±\sqrt{57}}{-4} when ± is plus. Add 3 to \sqrt{57}.

$$x=\frac{\sqrt{57}+3}{-4}$$

Divide 3+\sqrt{57} by -4.

$$x=\frac{-\sqrt{57}-3}{4}$$

Now solve the equation x=\frac{3±\sqrt{57}}{-4} when ± is minus. Subtract \sqrt{57} from 3.

$$x=\frac{3-\sqrt{57}}{-4}$$

Divide 3-\sqrt{57} by -4.

$$x=\frac{\sqrt{57}-3}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{57}-3}{4} x=\frac{\sqrt{57}-3}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x+3,x^{2}-9,3-x.

$$\left(x-3\right)\left(x+2\right)-x^{2}=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)\left(x-1\right)\right)$$

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

$$x^{2}-x-6-x^{2}=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)\left(x-1\right)\right)$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-\left(3+x\right)\left(x-1\right)\right)$$

Use the distributive property to multiply -1 by 3+x.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-3-x\right)\left(x-1\right)$$

Use the distributive property to multiply -3-x by x-1 and combine like terms.

$$x^{2}-x-6-x^{2}=x^{2}-9-\left(-2x+3-x^{2}\right)$$

To find the opposite of -2x+3-x^{2}, find the opposite of each term.

$$x^{2}-x-6-x^{2}=x^{2}-9+2x-3+x^{2}$$

Subtract 3 from -9 to get -12.

$$x^{2}-x-6-x^{2}=x^{2}-12+2x+x^{2}$$

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

$$x^{2}-x-6-x^{2}=2x^{2}-12+2x$$

Subtract 2x^{2} from both sides.

$$x^{2}-x-6-x^{2}-2x^{2}=-12+2x$$

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

$$-x^{2}-x-6-x^{2}=-12+2x$$

Subtract 2x from both sides.

$$-x^{2}-x-6-x^{2}-2x=-12$$

Combine -x and -2x to get -3x.

$$-x^{2}-3x-6-x^{2}=-12$$

Add 6 to both sides.

$$-x^{2}-3x-x^{2}=-12+6$$

Add -12 and 6 to get -6.

$$-x^{2}-3x-x^{2}=-6$$

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

$$-2x^{2}-3x=-6$$

Divide both sides by -2.

$$\frac{-2x^{2}-3x}{-2}=-\frac{6}{-2}$$

Dividing by -2 undoes the multiplication by -2.

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

Divide -3 by -2.

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

Divide -6 by -2.

$$x^{2}+\frac{3}{2}x=3$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=3+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=3+\frac{9}{16}$$

Add 3 to \frac{9}{16}.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{57}{16}$$

Factor x^{2}+\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{57}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{57}{16}}$$

Simplify.

$$x+\frac{3}{4}=\frac{\sqrt{57}}{4} x+\frac{3}{4}=-\frac{\sqrt{57}}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$x=\frac{\sqrt{57}-3}{4} x=\frac{-\sqrt{57}-3}{4}$$