Solve (x+2)*(2x-1)=(3x+1)*(x-1/2)

$$( x + 2 ) \cdot ( 2 x - 1 ) = ( 3 x + 1 ) \cdot ( x - \frac { 1 } { 2 } )$$

Solve for x

$$x=3$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

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

$$2x^{2}+3x-2=\left(3x+1\right)\left(x-\frac{1}{2}\right)$$

Use the distributive property to multiply 3x+1 by x-\frac{1}{2} and combine like terms.

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

Subtract 3x^{2} from both sides.

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

Combine 2x^{2} and -3x^{2} to get -x^{2}.

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

Add \frac{1}{2}x to both sides.

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

Combine 3x and \frac{1}{2}x to get \frac{7}{2}x.

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

Add \frac{1}{2} to both sides.

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

Add -2 and \frac{1}{2} to get -\frac{3}{2}.

$$-x^{2}+\frac{7}{2}x-\frac{3}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{7}{2} for b, and -\frac{3}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{7}{2}±\sqrt{\left(\frac{7}{2}\right)^{2}-4\left(-1\right)\left(-\frac{3}{2}\right)}}{2\left(-1\right)}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-4\left(-1\right)\left(-\frac{3}{2}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}+4\left(-\frac{3}{2}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{3}{2}.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-6}}{2\left(-1\right)}$$

Add \frac{49}{4} to -6.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{25}{4}}}{2\left(-1\right)}$$

Take the square root of \frac{25}{4}.

$$x=\frac{-\frac{7}{2}±\frac{5}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{7}{2}±\frac{5}{2}}{-2}$$

Now solve the equation x=\frac{-\frac{7}{2}±\frac{5}{2}}{-2} when ± is plus. Add -\frac{7}{2} to \frac{5}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -1 by -2.

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

Now solve the equation x=\frac{-\frac{7}{2}±\frac{5}{2}}{-2} when ± is minus. Subtract \frac{5}{2} from -\frac{7}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

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

Steps for Completing the Square

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

$$2x^{2}+3x-2=\left(3x+1\right)\left(x-\frac{1}{2}\right)$$

Use the distributive property to multiply 3x+1 by x-\frac{1}{2} and combine like terms.

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

Subtract 3x^{2} from both sides.

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

Combine 2x^{2} and -3x^{2} to get -x^{2}.

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

Add \frac{1}{2}x to both sides.

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

Combine 3x and \frac{1}{2}x to get \frac{7}{2}x.

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

Add 2 to both sides.

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

Add -\frac{1}{2} and 2 to get \frac{3}{2}.

$$-x^{2}+\frac{7}{2}x=\frac{3}{2}$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{7}{2}x}{-1}=\frac{\frac{3}{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{7}{2}}{-1}x=\frac{\frac{3}{2}}{-1}$$

Divide \frac{7}{2} by -1.

$$x^{2}-\frac{7}{2}x=\frac{\frac{3}{2}}{-1}$$

Divide \frac{3}{2} by -1.

$$x^{2}-\frac{7}{2}x=-\frac{3}{2}$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}$$

Add -\frac{3}{2} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{5}{4} x-\frac{7}{4}=-\frac{5}{4}$$

Add \frac{7}{4} to both sides of the equation.

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