Solve 1/2(x+8)(x)=42

$$\frac{ 1 }{ 2 } (x+8)(x)=42$$

Solve for x

$$x=-14$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2} by x+8.

$$\left(\frac{1}{2}x+\frac{1}{2}\times 8\right)x=42$$

Multiply \frac{1}{2} and 8 to get \frac{8}{2}.

$$\left(\frac{1}{2}x+\frac{8}{2}\right)x=42$$

Divide 8 by 2 to get 4.

$$\left(\frac{1}{2}x+4\right)x=42$$

Use the distributive property to multiply \frac{1}{2}x+4 by x.

$$\frac{1}{2}xx+4x=42$$

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

$$\frac{1}{2}x^{2}+4x=42$$

Subtract 42 from both sides.

$$\frac{1}{2}x^{2}+4x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 4 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times \frac{1}{2}\left(-42\right)}}{2\times \frac{1}{2}}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times \frac{1}{2}\left(-42\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-4±\sqrt{16-2\left(-42\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -42.

$$x=\frac{-4±\sqrt{16+84}}{2\times \frac{1}{2}}$$

Add 16 to 84.

$$x=\frac{-4±\sqrt{100}}{2\times \frac{1}{2}}$$

Take the square root of 100.

$$x=\frac{-4±10}{2\times \frac{1}{2}}$$

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

$$x=\frac{-4±10}{1}$$

Now solve the equation x=\frac{-4±10}{1} when ± is plus. Add -4 to 10.

$$x=\frac{6}{1}$$

Divide 6 by 1.

$$x=6$$

Now solve the equation x=\frac{-4±10}{1} when ± is minus. Subtract 10 from -4.

$$x=-\frac{14}{1}$$

Divide -14 by 1.

$$x=-14$$

The equation is now solved.

$$x=6 x=-14$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2} by x+8.

$$\left(\frac{1}{2}x+\frac{1}{2}\times 8\right)x=42$$

Multiply \frac{1}{2} and 8 to get \frac{8}{2}.

$$\left(\frac{1}{2}x+\frac{8}{2}\right)x=42$$

Divide 8 by 2 to get 4.

$$\left(\frac{1}{2}x+4\right)x=42$$

Use the distributive property to multiply \frac{1}{2}x+4 by x.

$$\frac{1}{2}xx+4x=42$$

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

$$\frac{1}{2}x^{2}+4x=42$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+4x}{\frac{1}{2}}=\frac{42}{\frac{1}{2}}$$

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

$$x^{2}+\frac{4}{\frac{1}{2}}x=\frac{42}{\frac{1}{2}}$$

Divide 4 by \frac{1}{2} by multiplying 4 by the reciprocal of \frac{1}{2}.

$$x^{2}+8x=\frac{42}{\frac{1}{2}}$$

Divide 42 by \frac{1}{2} by multiplying 42 by the reciprocal of \frac{1}{2}.

$$x^{2}+8x=84$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=84+4^{2}$$

Square 4.

$$x^{2}+8x+16=84+16$$

Add 84 to 16.

$$x^{2}+8x+16=100$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{100}$$

Simplify.

$$x+4=10 x+4=-10$$

Subtract 4 from both sides of the equation.

$$x=6 x=-14$$