Solve 1/2*(x+2)*x=40

$$\frac{ 1 }{ 2 } \times (x+2) \times x=40$$

Solve for x

$$x=-10$$

$$x=8$$

Steps Using the Quadratic Formula

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

$$\left(\frac{1}{2}x+\frac{1}{2}\times 2\right)x=40$$

Cancel out 2 and 2.

$$\left(\frac{1}{2}x+1\right)x=40$$

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

$$\frac{1}{2}xx+x=40$$

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

$$\frac{1}{2}x^{2}+x=40$$

Subtract 40 from both sides.

$$\frac{1}{2}x^{2}+x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 1 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{2}\left(-40\right)}}{2\times \frac{1}{2}}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times \frac{1}{2}\left(-40\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-1±\sqrt{1-2\left(-40\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -40.

$$x=\frac{-1±\sqrt{1+80}}{2\times \frac{1}{2}}$$

Add 1 to 80.

$$x=\frac{-1±\sqrt{81}}{2\times \frac{1}{2}}$$

Take the square root of 81.

$$x=\frac{-1±9}{2\times \frac{1}{2}}$$

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

$$x=\frac{-1±9}{1}$$

Now solve the equation x=\frac{-1±9}{1} when ± is plus. Add -1 to 9.

$$x=\frac{8}{1}$$

Divide 8 by 1.

$$x=8$$

Now solve the equation x=\frac{-1±9}{1} when ± is minus. Subtract 9 from -1.

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

Divide -10 by 1.

$$x=-10$$

The equation is now solved.

$$x=8 x=-10$$

Steps for Completing the Square

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

$$\left(\frac{1}{2}x+\frac{1}{2}\times 2\right)x=40$$

Cancel out 2 and 2.

$$\left(\frac{1}{2}x+1\right)x=40$$

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

$$\frac{1}{2}xx+x=40$$

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

$$\frac{1}{2}x^{2}+x=40$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+x}{\frac{1}{2}}=\frac{40}{\frac{1}{2}}$$

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

$$x^{2}+\frac{1}{\frac{1}{2}}x=\frac{40}{\frac{1}{2}}$$

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

$$x^{2}+2x=\frac{40}{\frac{1}{2}}$$

Divide 40 by \frac{1}{2} by multiplying 40 by the reciprocal of \frac{1}{2}.

$$x^{2}+2x=80$$

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

$$x^{2}+2x+1^{2}=80+1^{2}$$

Square 1.

$$x^{2}+2x+1=80+1$$

Add 80 to 1.

$$x^{2}+2x+1=81$$

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

$$\left(x+1\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{81}$$

Simplify.

$$x+1=9 x+1=-9$$

Subtract 1 from both sides of the equation.

$$x=8 x=-10$$