Solve 4-3x/x+3+5x-1/x+2=-5/2

$$\frac { 4 - 3 x } { x + 3 } + \frac { 5 x - 1 } { x + 2 } = - \frac { 5 } { 2 }$$

Solve for x

$$x = -\frac{40}{9} = -4\frac{4}{9} \approx -4.444444444$$

$$x=-1$$

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 2\left(x+2\right)\left(x+3\right), the least common multiple of x+3,x+2,2.

$$\left(2x+4\right)\left(4-3x\right)+\left(2x+6\right)\left(5x-1\right)=-5\left(x+2\right)\left(x+3\right)$$

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

$$-4x-6x^{2}+16+\left(2x+6\right)\left(5x-1\right)=-5\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+6 by 5x-1 and combine like terms.

$$-4x-6x^{2}+16+10x^{2}+28x-6=-5\left(x+2\right)\left(x+3\right)$$

Combine -6x^{2} and 10x^{2} to get 4x^{2}.

$$-4x+4x^{2}+16+28x-6=-5\left(x+2\right)\left(x+3\right)$$

Combine -4x and 28x to get 24x.

$$24x+4x^{2}+16-6=-5\left(x+2\right)\left(x+3\right)$$

Subtract 6 from 16 to get 10.

$$24x+4x^{2}+10=-5\left(x+2\right)\left(x+3\right)$$

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

$$24x+4x^{2}+10=\left(-5x-10\right)\left(x+3\right)$$

Use the distributive property to multiply -5x-10 by x+3 and combine like terms.

$$24x+4x^{2}+10=-5x^{2}-25x-30$$

Add 5x^{2} to both sides.

$$24x+4x^{2}+10+5x^{2}=-25x-30$$

Combine 4x^{2} and 5x^{2} to get 9x^{2}.

$$24x+9x^{2}+10=-25x-30$$

Add 25x to both sides.

$$24x+9x^{2}+10+25x=-30$$

Combine 24x and 25x to get 49x.

$$49x+9x^{2}+10=-30$$

Add 30 to both sides.

$$49x+9x^{2}+10+30=0$$

Add 10 and 30 to get 40.

$$49x+9x^{2}+40=0$$

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.

$$9x^{2}+49x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 49 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-49±\sqrt{49^{2}-4\times 9\times 40}}{2\times 9}$$

Square 49.

$$x=\frac{-49±\sqrt{2401-4\times 9\times 40}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-49±\sqrt{2401-36\times 40}}{2\times 9}$$

Multiply -36 times 40.

$$x=\frac{-49±\sqrt{2401-1440}}{2\times 9}$$

Add 2401 to -1440.

$$x=\frac{-49±\sqrt{961}}{2\times 9}$$

Take the square root of 961.

$$x=\frac{-49±31}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-49±31}{18}$$

Now solve the equation x=\frac{-49±31}{18} when ± is plus. Add -49 to 31.

$$x=-\frac{18}{18}$$

Divide -18 by 18.

$$x=-1$$

Now solve the equation x=\frac{-49±31}{18} when ± is minus. Subtract 31 from -49.

$$x=-\frac{80}{18}$$

Reduce the fraction \frac{-80}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{40}{9}$$

The equation is now solved.

$$x=-1 x=-\frac{40}{9}$$

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 2\left(x+2\right)\left(x+3\right), the least common multiple of x+3,x+2,2.

$$\left(2x+4\right)\left(4-3x\right)+\left(2x+6\right)\left(5x-1\right)=-5\left(x+2\right)\left(x+3\right)$$

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

$$-4x-6x^{2}+16+\left(2x+6\right)\left(5x-1\right)=-5\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 2x+6 by 5x-1 and combine like terms.

$$-4x-6x^{2}+16+10x^{2}+28x-6=-5\left(x+2\right)\left(x+3\right)$$

Combine -6x^{2} and 10x^{2} to get 4x^{2}.

$$-4x+4x^{2}+16+28x-6=-5\left(x+2\right)\left(x+3\right)$$

Combine -4x and 28x to get 24x.

$$24x+4x^{2}+16-6=-5\left(x+2\right)\left(x+3\right)$$

Subtract 6 from 16 to get 10.

$$24x+4x^{2}+10=-5\left(x+2\right)\left(x+3\right)$$

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

$$24x+4x^{2}+10=\left(-5x-10\right)\left(x+3\right)$$

Use the distributive property to multiply -5x-10 by x+3 and combine like terms.

$$24x+4x^{2}+10=-5x^{2}-25x-30$$

Add 5x^{2} to both sides.

$$24x+4x^{2}+10+5x^{2}=-25x-30$$

Combine 4x^{2} and 5x^{2} to get 9x^{2}.

$$24x+9x^{2}+10=-25x-30$$

Add 25x to both sides.

$$24x+9x^{2}+10+25x=-30$$

Combine 24x and 25x to get 49x.

$$49x+9x^{2}+10=-30$$

Subtract 10 from both sides.

$$49x+9x^{2}=-30-10$$

Subtract 10 from -30 to get -40.

$$49x+9x^{2}=-40$$

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.

$$9x^{2}+49x=-40$$

Divide both sides by 9.

$$\frac{9x^{2}+49x}{9}=-\frac{40}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{49}{9}x=-\frac{40}{9}$$

Divide \frac{49}{9}, the coefficient of the x term, by 2 to get \frac{49}{18}. Then add the square of \frac{49}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{49}{9}x+\left(\frac{49}{18}\right)^{2}=-\frac{40}{9}+\left(\frac{49}{18}\right)^{2}$$

Square \frac{49}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{49}{9}x+\frac{2401}{324}=-\frac{40}{9}+\frac{2401}{324}$$

Add -\frac{40}{9} to \frac{2401}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{49}{9}x+\frac{2401}{324}=\frac{961}{324}$$

Factor x^{2}+\frac{49}{9}x+\frac{2401}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{49}{18}\right)^{2}=\frac{961}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{49}{18}\right)^{2}}=\sqrt{\frac{961}{324}}$$

Simplify.

$$x+\frac{49}{18}=\frac{31}{18} x+\frac{49}{18}=-\frac{31}{18}$$

Subtract \frac{49}{18} from both sides of the equation.

$$x=-1 x=-\frac{40}{9}$$