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

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

Solve for x

$$x = \frac{\sqrt{105} - 1}{4} \approx 2.311737691$$

$$x=\frac{-\sqrt{105}-1}{4}\approx -2.811737691$$

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.

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

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

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

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

$$2x-3=5x+10-2x^{2}-4x$$

Combine 5x and -4x to get x.

$$2x-3=x+10-2x^{2}$$

Subtract x from both sides.

$$2x-3-x=10-2x^{2}$$

Combine 2x and -x to get x.

$$x-3=10-2x^{2}$$

Subtract 10 from both sides.

$$x-3-10=-2x^{2}$$

Subtract 10 from -3 to get -13.

$$x-13=-2x^{2}$$

Add 2x^{2} to both sides.

$$x-13+2x^{2}=0$$

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.

$$2x^{2}+x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-13\right)}}{2\times 2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-13\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8\left(-13\right)}}{2\times 2}$$

Multiply -8 times -13.

$$x=\frac{-1±\sqrt{1+104}}{2\times 2}$$

Add 1 to 104.

$$x=\frac{-1±\sqrt{105}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-1±\sqrt{105}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{105}}{4} when ± is plus. Add -1 to \sqrt{105}.

$$x=\frac{\sqrt{105}-1}{4}$$

Now solve the equation x=\frac{-1±\sqrt{105}}{4} when ± is minus. Subtract \sqrt{105} from -1.

$$x=\frac{-\sqrt{105}-1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{105}-1}{4} x=\frac{-\sqrt{105}-1}{4}$$

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.

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

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

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

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

$$2x-3=5x+10-2x^{2}-4x$$

Combine 5x and -4x to get x.

$$2x-3=x+10-2x^{2}$$

Subtract x from both sides.

$$2x-3-x=10-2x^{2}$$

Combine 2x and -x to get x.

$$x-3=10-2x^{2}$$

Add 2x^{2} to both sides.

$$x-3+2x^{2}=10$$

Add 3 to both sides.

$$x+2x^{2}=10+3$$

Add 10 and 3 to get 13.

$$x+2x^{2}=13$$

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.

$$2x^{2}+x=13$$

Divide both sides by 2.

$$\frac{2x^{2}+x}{2}=\frac{13}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{1}{2}x=\frac{13}{2}$$

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.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{13}{2}+\left(\frac{1}{4}\right)^{2}$$

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

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{13}{2}+\frac{1}{16}$$

Add \frac{13}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{105}{16}$$

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}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{105}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{105}}{4} x+\frac{1}{4}=-\frac{\sqrt{105}}{4}$$

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

$$x=\frac{\sqrt{105}-1}{4} x=\frac{-\sqrt{105}-1}{4}$$