Solve x^2-(7+x/2)((7+x/2)+x)=11

$${ x }^{ 2 } -( \frac{ 7+x }{ 2 } )(( \frac{ 7+x }{ 2 } )+x)=11$$

Solve for x

$$x=-3$$

$$x=31$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

Use the distributive property to multiply 7+x by \frac{7+x}{2}+x.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Express 7\times \frac{7+x}{2} as a single fraction.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Express x\times \frac{7+x}{2} as a single fraction.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

Since \frac{7\left(7+x\right)}{2} and \frac{x\left(7+x\right)}{2} have the same denominator, add them by adding their numerators.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

Do the multiplications in 7\left(7+x\right)+x\left(7+x\right).

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

Combine like terms in 49+7x+7x+x^{2}.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

To find the opposite of \frac{49+14x+x^{2}}{2}+7x+x^{2}, find the opposite of each term.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

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

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

Divide each term of 49+14x+x^{2} by 2 to get \frac{49}{2}+7x+\frac{1}{2}x^{2}.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

To find the opposite of \frac{49}{2}+7x+\frac{1}{2}x^{2}, find the opposite of each term.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

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

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

Combine -7x and -7x to get -14x.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

Subtract 22 from both sides.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x-22=0$$

Subtract 22 from -\frac{49}{2} to get -\frac{93}{2}.

$$\frac{1}{2}x^{2}-\frac{93}{2}-14x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{1}{2}x^{2}-14x-\frac{93}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -14 for b, and -\frac{93}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-14\right)±\sqrt{196-2\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{93}{2}.

$$x=\frac{-\left(-14\right)±\sqrt{196+93}}{2\times \frac{1}{2}}$$

Add 196 to 93.

$$x=\frac{-\left(-14\right)±\sqrt{289}}{2\times \frac{1}{2}}$$

Take the square root of 289.

$$x=\frac{-\left(-14\right)±17}{2\times \frac{1}{2}}$$

The opposite of -14 is 14.

$$x=\frac{14±17}{2\times \frac{1}{2}}$$

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

$$x=\frac{14±17}{1}$$

Now solve the equation x=\frac{14±17}{1} when ± is plus. Add 14 to 17.

$$x=\frac{31}{1}$$

Divide 31 by 1.

$$x=31$$

Now solve the equation x=\frac{14±17}{1} when ± is minus. Subtract 17 from 14.

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

Divide -3 by 1.

$$x=-3$$

The equation is now solved.

$$x=31 x=-3$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

Use the distributive property to multiply 7+x by \frac{7+x}{2}+x.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Express 7\times \frac{7+x}{2} as a single fraction.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

Express x\times \frac{7+x}{2} as a single fraction.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

Since \frac{7\left(7+x\right)}{2} and \frac{x\left(7+x\right)}{2} have the same denominator, add them by adding their numerators.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

Do the multiplications in 7\left(7+x\right)+x\left(7+x\right).

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

Combine like terms in 49+7x+7x+x^{2}.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

To find the opposite of \frac{49+14x+x^{2}}{2}+7x+x^{2}, find the opposite of each term.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

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

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

Divide each term of 49+14x+x^{2} by 2 to get \frac{49}{2}+7x+\frac{1}{2}x^{2}.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

To find the opposite of \frac{49}{2}+7x+\frac{1}{2}x^{2}, find the opposite of each term.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

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

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

Combine -7x and -7x to get -14x.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

Add \frac{49}{2} to both sides.

$$\frac{1}{2}x^{2}-14x=22+\frac{49}{2}$$

Add 22 and \frac{49}{2} to get \frac{93}{2}.

$$\frac{1}{2}x^{2}-14x=\frac{93}{2}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-14x}{\frac{1}{2}}=\frac{\frac{93}{2}}{\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{14}{\frac{1}{2}}\right)x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

Divide -14 by \frac{1}{2} by multiplying -14 by the reciprocal of \frac{1}{2}.

$$x^{2}-28x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

Divide \frac{93}{2} by \frac{1}{2} by multiplying \frac{93}{2} by the reciprocal of \frac{1}{2}.

$$x^{2}-28x=93$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-28x+\left(-14\right)^{2}=93+\left(-14\right)^{2}$$

Square -14.

$$x^{2}-28x+196=93+196$$

Add 93 to 196.

$$x^{2}-28x+196=289$$

Factor x^{2}-28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-14\right)^{2}=289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{289}$$

Simplify.

$$x-14=17 x-14=-17$$

Add 14 to both sides of the equation.

$$x=31 x=-3$$