Solve 3x-7/x+5=x-3/x+2

$$\frac { 3 x - 7 } { x + 5 } = \frac { x - 3 } { x + 2 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+5\right), the least common multiple of x+5,x+2.

$$\left(x+2\right)\left(3x-7\right)=\left(x+5\right)\left(x-3\right)$$

Use the distributive property to multiply x+2 by 3x-7 and combine like terms.

$$3x^{2}-x-14=\left(x+5\right)\left(x-3\right)$$

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

$$3x^{2}-x-14=x^{2}+2x-15$$

Subtract x^{2} from both sides.

$$3x^{2}-x-14-x^{2}=2x-15$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}-x-14=2x-15$$

Subtract 2x from both sides.

$$2x^{2}-x-14-2x=-15$$

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

$$2x^{2}-3x-14=-15$$

Add 15 to both sides.

$$2x^{2}-3x-14+15=0$$

Add -14 and 15 to get 1.

$$2x^{2}-3x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -3.

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

Multiply -4 times 2.

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

Add 9 to -8.

$$x=\frac{-\left(-3\right)±\sqrt{1}}{2\times 2}$$

Take the square root of 1.

$$x=\frac{-\left(-3\right)±1}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±1}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±1}{4}$$

Now solve the equation x=\frac{3±1}{4} when ± is plus. Add 3 to 1.

$$x=\frac{4}{4}$$

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{3±1}{4} when ± is minus. Subtract 1 from 3.

$$x=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=1 x=\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+5\right), the least common multiple of x+5,x+2.

$$\left(x+2\right)\left(3x-7\right)=\left(x+5\right)\left(x-3\right)$$

Use the distributive property to multiply x+2 by 3x-7 and combine like terms.

$$3x^{2}-x-14=\left(x+5\right)\left(x-3\right)$$

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

$$3x^{2}-x-14=x^{2}+2x-15$$

Subtract x^{2} from both sides.

$$3x^{2}-x-14-x^{2}=2x-15$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}-x-14=2x-15$$

Subtract 2x from both sides.

$$2x^{2}-x-14-2x=-15$$

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

$$2x^{2}-3x-14=-15$$

Add 14 to both sides.

$$2x^{2}-3x=-15+14$$

Add -15 and 14 to get -1.

$$2x^{2}-3x=-1$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

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.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{3}{4}\right)^{2}$$

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

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}$$

Add -\frac{1}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{16}$$

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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{1}{4} x-\frac{3}{4}=-\frac{1}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=1 x=\frac{1}{2}$$