Solve (x-3)/(x+2)=(-x-0,5)

$$\frac { ( x - 3 ) } { ( x + 2 ) } = ( - x - 0,5 )$$

Solve for x

$$x=-4$$

$$x=0,5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$x-3=-\left(x+2\right)x+\left(x+2\right)\left(-0,5\right)$$

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

$$x-3=-\left(x^{2}+2x\right)+\left(x+2\right)\left(-0,5\right)$$

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

$$x-3=-x^{2}-2x+\left(x+2\right)\left(-0,5\right)$$

Use the distributive property to multiply x+2 by -0,5.

$$x-3=-x^{2}-2x-0,5x-1$$

Combine -2x and -0,5x to get -2,5x.

$$x-3=-x^{2}-2,5x-1$$

Add x^{2} to both sides.

$$x-3+x^{2}=-2,5x-1$$

Add 2,5x to both sides.

$$x-3+x^{2}+2,5x=-1$$

Combine x and 2,5x to get 3,5x.

$$3,5x-3+x^{2}=-1$$

Add 1 to both sides.

$$3,5x-3+x^{2}+1=0$$

Add -3 and 1 to get -2.

$$3,5x-2+x^{2}=0$$

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.

$$x^{2}+3,5x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3,5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3,5±\sqrt{3,5^{2}-4\left(-2\right)}}{2}$$

Square 3,5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3,5±\sqrt{12,25-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-3,5±\sqrt{12,25+8}}{2}$$

Add 12,25 to 8.

$$x=\frac{-3,5±\sqrt{20,25}}{2}$$

Take the square root of 20,25.

$$x=\frac{-3,5±\frac{9}{2}}{2}$$

Now solve the equation x=\frac{-3,5±\frac{9}{2}}{2} when ± is plus. Add -3,5 to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Now solve the equation x=\frac{-3,5±\frac{9}{2}}{2} when ± is minus. Subtract \frac{9}{2} from -3,5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=\frac{1}{2} x=-4$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$x-3=-\left(x+2\right)x+\left(x+2\right)\left(-0,5\right)$$

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

$$x-3=-\left(x^{2}+2x\right)+\left(x+2\right)\left(-0,5\right)$$

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

$$x-3=-x^{2}-2x+\left(x+2\right)\left(-0,5\right)$$

Use the distributive property to multiply x+2 by -0,5.

$$x-3=-x^{2}-2x-0,5x-1$$

Combine -2x and -0,5x to get -2,5x.

$$x-3=-x^{2}-2,5x-1$$

Add x^{2} to both sides.

$$x-3+x^{2}=-2,5x-1$$

Add 2,5x to both sides.

$$x-3+x^{2}+2,5x=-1$$

Combine x and 2,5x to get 3,5x.

$$3,5x-3+x^{2}=-1$$

Add 3 to both sides.

$$3,5x+x^{2}=-1+3$$

Add -1 and 3 to get 2.

$$3,5x+x^{2}=2$$

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.

$$x^{2}+3,5x=2$$

Divide 3,5, the coefficient of the x term, by 2 to get 1,75. Then add the square of 1,75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3,5x+1,75^{2}=2+1,75^{2}$$

Square 1,75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3,5x+3,0625=2+3,0625$$

Add 2 to 3,0625.

$$x^{2}+3,5x+3,0625=5,0625$$

Factor x^{2}+3,5x+3,0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1,75\right)^{2}=5,0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1,75\right)^{2}}=\sqrt{5,0625}$$

Simplify.

$$x+1,75=\frac{9}{4} x+1,75=-\frac{9}{4}$$

Subtract 1,75 from both sides of the equation.

$$x=\frac{1}{2} x=-4$$