Solve 1/2[x+(x+1.4)](x-0.5)=4.05

$$\frac { 1 } { 2 } [ x + ( x + 1.4 ) ] ( x - 0.5 ) = 4.05$$

Solve for x

$$x=-2.2$$

$$x=2$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$\frac{1}{2}\left(2x+1.4\right)\left(x-0.5\right)=4.05$$

Use the distributive property to multiply \frac{1}{2} by 2x+1.4.

$$\left(x+\frac{7}{10}\right)\left(x-0.5\right)=4.05$$

Use the distributive property to multiply x+\frac{7}{10} by x-0.5 and combine like terms.

$$x^{2}+\frac{1}{5}x-\frac{7}{20}=4.05$$

Subtract 4.05 from both sides.

$$x^{2}+\frac{1}{5}x-\frac{7}{20}-4.05=0$$

Subtract 4.05 from -\frac{7}{20} to get -\frac{22}{5}.

$$x^{2}+\frac{1}{5}x-\frac{22}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{1}{5} for b, and -\frac{22}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{5}±\sqrt{\left(\frac{1}{5}\right)^{2}-4\left(-\frac{22}{5}\right)}}{2}$$

Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{1}{25}-4\left(-\frac{22}{5}\right)}}{2}$$

Multiply -4 times -\frac{22}{5}.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{1}{25}+\frac{88}{5}}}{2}$$

Add \frac{1}{25} to \frac{88}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{441}{25}}}{2}$$

Take the square root of \frac{441}{25}.

$$x=\frac{-\frac{1}{5}±\frac{21}{5}}{2}$$

Now solve the equation x=\frac{-\frac{1}{5}±\frac{21}{5}}{2} when ± is plus. Add -\frac{1}{5} to \frac{21}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-\frac{1}{5}±\frac{21}{5}}{2} when ± is minus. Subtract \frac{21}{5} from -\frac{1}{5} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{22}{5}}{2}$$

Divide -\frac{22}{5} by 2.

$$x=-\frac{11}{5}$$

The equation is now solved.

$$x=2 x=-\frac{11}{5}$$

Steps for Completing the Square

Combine x and x to get 2x.

$$\frac{1}{2}\left(2x+1.4\right)\left(x-0.5\right)=4.05$$

Use the distributive property to multiply \frac{1}{2} by 2x+1.4.

$$\left(x+\frac{7}{10}\right)\left(x-0.5\right)=4.05$$

Use the distributive property to multiply x+\frac{7}{10} by x-0.5 and combine like terms.

$$x^{2}+\frac{1}{5}x-\frac{7}{20}=4.05$$

Add \frac{7}{20} to both sides.

$$x^{2}+\frac{1}{5}x=4.05+\frac{7}{20}$$

Add 4.05 and \frac{7}{20} to get \frac{22}{5}.

$$x^{2}+\frac{1}{5}x=\frac{22}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\frac{22}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{22}{5}+\frac{1}{100}$$

Add \frac{22}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{441}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{441}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{441}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{21}{10} x+\frac{1}{10}=-\frac{21}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=2 x=-\frac{11}{5}$$