Solve 5/x+3+4/x+2=2

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

Solve for x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$\left(x+2\right)\times 5+\left(x+3\right)\times 4=2\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+2 by 5.

$$5x+10+\left(x+3\right)\times 4=2\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 4.

$$5x+10+4x+12=2\left(x+2\right)\left(x+3\right)$$

Combine 5x and 4x to get 9x.

$$9x+10+12=2\left(x+2\right)\left(x+3\right)$$

Add 10 and 12 to get 22.

$$9x+22=2\left(x+2\right)\left(x+3\right)$$

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

$$9x+22=\left(2x+4\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+4 by x+3 and combine like terms.

$$9x+22=2x^{2}+10x+12$$

Subtract 2x^{2} from both sides.

$$9x+22-2x^{2}=10x+12$$

Subtract 10x from both sides.

$$9x+22-2x^{2}-10x=12$$

Combine 9x and -10x to get -x.

$$-x+22-2x^{2}=12$$

Subtract 12 from both sides.

$$-x+22-2x^{2}-12=0$$

Subtract 12 from 22 to get 10.

$$-x+10-2x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}-x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -1 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-2\right)\times 10}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-1\right)±\sqrt{1+8\times 10}}{2\left(-2\right)}$$

Multiply 8 times 10.

$$x=\frac{-\left(-1\right)±\sqrt{1+80}}{2\left(-2\right)}$$

Add 1 to 80.

$$x=\frac{-\left(-1\right)±\sqrt{81}}{2\left(-2\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-1\right)±9}{2\left(-2\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±9}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{1±9}{-4}$$

Now solve the equation x=\frac{1±9}{-4} when ± is plus. Add 1 to 9.

$$x=\frac{10}{-4}$$

Reduce the fraction \frac{10}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{2}$$

Now solve the equation x=\frac{1±9}{-4} when ± is minus. Subtract 9 from 1.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

The equation is now solved.

$$x=-\frac{5}{2} x=2$$

Steps for Completing the Square

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

$$\left(x+2\right)\times 5+\left(x+3\right)\times 4=2\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+2 by 5.

$$5x+10+\left(x+3\right)\times 4=2\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 4.

$$5x+10+4x+12=2\left(x+2\right)\left(x+3\right)$$

Combine 5x and 4x to get 9x.

$$9x+10+12=2\left(x+2\right)\left(x+3\right)$$

Add 10 and 12 to get 22.

$$9x+22=2\left(x+2\right)\left(x+3\right)$$

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

$$9x+22=\left(2x+4\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+4 by x+3 and combine like terms.

$$9x+22=2x^{2}+10x+12$$

Subtract 2x^{2} from both sides.

$$9x+22-2x^{2}=10x+12$$

Subtract 10x from both sides.

$$9x+22-2x^{2}-10x=12$$

Combine 9x and -10x to get -x.

$$-x+22-2x^{2}=12$$

Subtract 22 from both sides.

$$-x-2x^{2}=12-22$$

Subtract 22 from 12 to get -10.

$$-x-2x^{2}=-10$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}-x=-10$$

Divide both sides by -2.

$$\frac{-2x^{2}-x}{-2}=-\frac{10}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{1}{-2}\right)x=-\frac{10}{-2}$$

Divide -1 by -2.

$$x^{2}+\frac{1}{2}x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x^{2}+\frac{1}{2}x=5$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=5+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=5+\frac{1}{16}$$

Add 5 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{81}{16}$$

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

$$\left(x+\frac{1}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

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

Subtract \frac{1}{4} from both sides of the equation.

$$x=2 x=-\frac{5}{2}$$