Solve 0.5(2+x)(1+x)=3x

$$0.5(2+x)(1+x)=3x$$

Solve for x

$$x=1$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.5 by 2+x.

$$\left(1+0.5x\right)\left(1+x\right)=3x$$

Use the distributive property to multiply 1+0.5x by 1+x and combine like terms.

$$1+1.5x+0.5x^{2}=3x$$

Subtract 3x from both sides.

$$1+1.5x+0.5x^{2}-3x=0$$

Combine 1.5x and -3x to get -1.5x.

$$1-1.5x+0.5x^{2}=0$$

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.

$$0.5x^{2}-1.5x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.5 for a, -1.5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.5\right)±\sqrt{\left(-1.5\right)^{2}-4\times 0.5}}{2\times 0.5}$$

Square -1.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.5\right)±\sqrt{2.25-4\times 0.5}}{2\times 0.5}$$

Multiply -4 times 0.5.

$$x=\frac{-\left(-1.5\right)±\sqrt{2.25-2}}{2\times 0.5}$$

Add 2.25 to -2.

$$x=\frac{-\left(-1.5\right)±\sqrt{0.25}}{2\times 0.5}$$

Take the square root of 0.25.

$$x=\frac{-\left(-1.5\right)±\frac{1}{2}}{2\times 0.5}$$

The opposite of -1.5 is 1.5.

$$x=\frac{1.5±\frac{1}{2}}{2\times 0.5}$$

Multiply 2 times 0.5.

$$x=\frac{1.5±\frac{1}{2}}{1}$$

Now solve the equation x=\frac{1.5±\frac{1}{2}}{1} when ± is plus. Add 1.5 to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 2 by 1.

$$x=2$$

Now solve the equation x=\frac{1.5±\frac{1}{2}}{1} when ± is minus. Subtract \frac{1}{2} from 1.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 1 by 1.

$$x=1$$

The equation is now solved.

$$x=2 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 0.5 by 2+x.

$$\left(1+0.5x\right)\left(1+x\right)=3x$$

Use the distributive property to multiply 1+0.5x by 1+x and combine like terms.

$$1+1.5x+0.5x^{2}=3x$$

Subtract 3x from both sides.

$$1+1.5x+0.5x^{2}-3x=0$$

Combine 1.5x and -3x to get -1.5x.

$$1-1.5x+0.5x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-1.5x+0.5x^{2}=-1$$

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.

$$0.5x^{2}-1.5x=-1$$

Multiply both sides by 2.

$$\frac{0.5x^{2}-1.5x}{0.5}=-\frac{1}{0.5}$$

Dividing by 0.5 undoes the multiplication by 0.5.

$$x^{2}+\left(-\frac{1.5}{0.5}\right)x=-\frac{1}{0.5}$$

Divide -1.5 by 0.5 by multiplying -1.5 by the reciprocal of 0.5.

$$x^{2}-3x=-\frac{1}{0.5}$$

Divide -1 by 0.5 by multiplying -1 by the reciprocal of 0.5.

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

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-2+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+2.25=-2+2.25$$

Add -2 to 2.25.

$$x^{2}-3x+2.25=0.25$$

Factor x^{2}-3x+2.25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=0.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{0.25}$$

Simplify.

$$x-\frac{3}{2}=\frac{1}{2} x-\frac{3}{2}=-\frac{1}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=2 x=1$$