Solve 4/x+3=2x+3/x+2-6

$$\frac{ 4 }{ x+3 } = \frac{ 2x+3 }{ x+2 } -6$$

Solve for x

$$x=\frac{\sqrt{65}-25}{8}\approx -2.117217781$$

$$x=\frac{-\sqrt{65}-25}{8}\approx -4.132782219$$

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.

$$\left(x+2\right)\times 4=\left(x+3\right)\left(2x+3\right)+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

Use the distributive property to multiply x+2 by 4.

$$4x+8=\left(x+3\right)\left(2x+3\right)+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

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

$$4x+8=2x^{2}+9x+9+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

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

$$4x+8=2x^{2}+9x+9+\left(x^{2}+5x+6\right)\left(-6\right)$$

Use the distributive property to multiply x^{2}+5x+6 by -6.

$$4x+8=2x^{2}+9x+9-6x^{2}-30x-36$$

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

$$4x+8=-4x^{2}+9x+9-30x-36$$

Combine 9x and -30x to get -21x.

$$4x+8=-4x^{2}-21x+9-36$$

Subtract 36 from 9 to get -27.

$$4x+8=-4x^{2}-21x-27$$

Add 4x^{2} to both sides.

$$4x+8+4x^{2}=-21x-27$$

Add 21x to both sides.

$$4x+8+4x^{2}+21x=-27$$

Combine 4x and 21x to get 25x.

$$25x+8+4x^{2}=-27$$

Add 27 to both sides.

$$25x+8+4x^{2}+27=0$$

Add 8 and 27 to get 35.

$$25x+35+4x^{2}=0$$

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.

$$4x^{2}+25x+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 25 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 4\times 35}}{2\times 4}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 4\times 35}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-25±\sqrt{625-16\times 35}}{2\times 4}$$

Multiply -16 times 35.

$$x=\frac{-25±\sqrt{625-560}}{2\times 4}$$

Add 625 to -560.

$$x=\frac{-25±\sqrt{65}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-25±\sqrt{65}}{8}$$

Now solve the equation x=\frac{-25±\sqrt{65}}{8} when ± is plus. Add -25 to \sqrt{65}.

$$x=\frac{\sqrt{65}-25}{8}$$

Now solve the equation x=\frac{-25±\sqrt{65}}{8} when ± is minus. Subtract \sqrt{65} from -25.

$$x=\frac{-\sqrt{65}-25}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{65}-25}{8} x=\frac{-\sqrt{65}-25}{8}$$

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.

$$\left(x+2\right)\times 4=\left(x+3\right)\left(2x+3\right)+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

Use the distributive property to multiply x+2 by 4.

$$4x+8=\left(x+3\right)\left(2x+3\right)+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

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

$$4x+8=2x^{2}+9x+9+\left(x+2\right)\left(x+3\right)\left(-6\right)$$

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

$$4x+8=2x^{2}+9x+9+\left(x^{2}+5x+6\right)\left(-6\right)$$

Use the distributive property to multiply x^{2}+5x+6 by -6.

$$4x+8=2x^{2}+9x+9-6x^{2}-30x-36$$

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

$$4x+8=-4x^{2}+9x+9-30x-36$$

Combine 9x and -30x to get -21x.

$$4x+8=-4x^{2}-21x+9-36$$

Subtract 36 from 9 to get -27.

$$4x+8=-4x^{2}-21x-27$$

Add 4x^{2} to both sides.

$$4x+8+4x^{2}=-21x-27$$

Add 21x to both sides.

$$4x+8+4x^{2}+21x=-27$$

Combine 4x and 21x to get 25x.

$$25x+8+4x^{2}=-27$$

Subtract 8 from both sides.

$$25x+4x^{2}=-27-8$$

Subtract 8 from -27 to get -35.

$$25x+4x^{2}=-35$$

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.

$$4x^{2}+25x=-35$$

Divide both sides by 4.

$$\frac{4x^{2}+25x}{4}=-\frac{35}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{25}{4}x=-\frac{35}{4}$$

Divide \frac{25}{4}, the coefficient of the x term, by 2 to get \frac{25}{8}. Then add the square of \frac{25}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{4}x+\left(\frac{25}{8}\right)^{2}=-\frac{35}{4}+\left(\frac{25}{8}\right)^{2}$$

Square \frac{25}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{4}x+\frac{625}{64}=-\frac{35}{4}+\frac{625}{64}$$

Add -\frac{35}{4} to \frac{625}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{4}x+\frac{625}{64}=\frac{65}{64}$$

Factor x^{2}+\frac{25}{4}x+\frac{625}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{8}\right)^{2}=\frac{65}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{8}\right)^{2}}=\sqrt{\frac{65}{64}}$$

Simplify.

$$x+\frac{25}{8}=\frac{\sqrt{65}}{8} x+\frac{25}{8}=-\frac{\sqrt{65}}{8}$$

Subtract \frac{25}{8} from both sides of the equation.

$$x=\frac{\sqrt{65}-25}{8} x=\frac{-\sqrt{65}-25}{8}$$