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

$$\frac { 1 } { 2 } ( x + 1 ) ( x + 2 ) = x \times x$$

Solve for x

$$x = \frac{\sqrt{17} + 3}{2} \approx 3.561552813$$

$$x=\frac{3-\sqrt{17}}{2}\approx -0.561552813$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

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

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

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

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

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

Subtract x^{2} from both sides.

$$\frac{1}{2}x^{2}+\frac{3}{2}x+1-x^{2}=0$$

Combine \frac{1}{2}x^{2} and -x^{2} to get -\frac{1}{2}x^{2}.

$$-\frac{1}{2}x^{2}+\frac{3}{2}x+1=0$$

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

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

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

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}-4\left(-\frac{1}{2}\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply -4 times -\frac{1}{2}.

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}+2}}{2\left(-\frac{1}{2}\right)}$$

Add \frac{9}{4} to 2.

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{17}{4}}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of \frac{17}{4}.

$$x=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -\frac{1}{2}.

$$x=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1}$$

Now solve the equation x=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1} when ± is plus. Add -\frac{3}{2} to \frac{\sqrt{17}}{2}.

$$x=\frac{\sqrt{17}-3}{-2}$$

Divide \frac{-3+\sqrt{17}}{2} by -1.

$$x=\frac{3-\sqrt{17}}{2}$$

Now solve the equation x=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1} when ± is minus. Subtract \frac{\sqrt{17}}{2} from -\frac{3}{2}.

$$x=\frac{-\sqrt{17}-3}{-2}$$

Divide \frac{-3-\sqrt{17}}{2} by -1.

$$x=\frac{\sqrt{17}+3}{2}$$

The equation is now solved.

$$x=\frac{3-\sqrt{17}}{2} x=\frac{\sqrt{17}+3}{2}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

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

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

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

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

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

Subtract x^{2} from both sides.

$$\frac{1}{2}x^{2}+\frac{3}{2}x+1-x^{2}=0$$

Combine \frac{1}{2}x^{2} and -x^{2} to get -\frac{1}{2}x^{2}.

$$-\frac{1}{2}x^{2}+\frac{3}{2}x+1=0$$

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

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

Multiply both sides by -2.

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

Dividing by -\frac{1}{2} undoes the multiplication by -\frac{1}{2}.

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

Divide \frac{3}{2} by -\frac{1}{2} by multiplying \frac{3}{2} by the reciprocal of -\frac{1}{2}.

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

Divide -1 by -\frac{1}{2} by multiplying -1 by the reciprocal of -\frac{1}{2}.

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

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.

$$x^{2}-3x+\frac{9}{4}=2+\frac{9}{4}$$

Add 2 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{17}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{17}}{2} x-\frac{3}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}+3}{2} x=\frac{3-\sqrt{17}}{2}$$