Solve 1/2[x+(x+14)](x-05)=405

$$\frac { 1 } { 2 } [ x + ( x + 14 ) ] ( x - 05 ) = 405$$

Solve for x

$$x = \frac{\sqrt{1669} - 7}{2} \approx 16.926698216$$

$$x=\frac{-\sqrt{1669}-7}{2}\approx -23.926698216$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$\frac{1}{2}\left(2x+14\right)\left(x-0\times 5\right)=405$$

Multiply 0 and 5 to get 0.

$$\frac{1}{2}\left(2x+14\right)\left(x-0\right)=405$$

Use the distributive property to multiply \frac{1}{2} by 2x+14.

$$\left(x+7\right)\left(x-0\right)=405$$

Use the distributive property to multiply x+7 by x-0.

$$x\left(x-0\right)+7\left(x-0\right)=405$$

Subtract 405 from both sides.

$$x\left(x-0\right)+7\left(x-0\right)-405=0$$

Reorder the terms.

$$xx+7x-405=0$$

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

$$x^{2}+7x-405=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -405 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-405\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-405\right)}}{2}$$

Multiply -4 times -405.

$$x=\frac{-7±\sqrt{49+1620}}{2}$$

Add 49 to 1620.

$$x=\frac{-7±\sqrt{1669}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{1669}}{2} when ± is plus. Add -7 to \sqrt{1669}.

$$x=\frac{\sqrt{1669}-7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{1669}}{2} when ± is minus. Subtract \sqrt{1669} from -7.

$$x=\frac{-\sqrt{1669}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1669}-7}{2} x=\frac{-\sqrt{1669}-7}{2}$$

Steps for Completing the Square

Combine x and x to get 2x.

$$\frac{1}{2}\left(2x+14\right)\left(x-0\times 5\right)=405$$

Multiply 0 and 5 to get 0.

$$\frac{1}{2}\left(2x+14\right)\left(x-0\right)=405$$

Use the distributive property to multiply \frac{1}{2} by 2x+14.

$$\left(x+7\right)\left(x-0\right)=405$$

Use the distributive property to multiply x+7 by x-0.

$$x\left(x-0\right)+7\left(x-0\right)=405$$

Reorder the terms.

$$xx+7x=405$$

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

$$x^{2}+7x=405$$

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

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=405+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=405+\frac{49}{4}$$

Add 405 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{1669}{4}$$

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

$$\left(x+\frac{7}{2}\right)^{2}=\frac{1669}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{1669}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{1669}}{2} x+\frac{7}{2}=-\frac{\sqrt{1669}}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{\sqrt{1669}-7}{2} x=\frac{-\sqrt{1669}-7}{2}$$