Solve (1/2(2x+2)(x+4))-(1/2(x+1)(x))=60

$$( \frac{ 1 }{ 2 } (2x+2)(x+4))-( \frac{ 1 }{ 2 } (x+1)(x))=60$$

Solve for x

$$x=-16$$

$$x=7$$

Steps Using the Quadratic Formula

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

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

Use the distributive property to multiply x+1 by x+4 and combine like terms.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

Subtract 60 from both sides.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x-60=0$$

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

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x-60=0$$

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

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x-60=0$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x-60=0$$

Combine 5x and -\frac{1}{2}x to get \frac{9}{2}x.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4-60=0$$

Subtract 60 from 4 to get -56.

$$\frac{1}{2}x^{2}+\frac{9}{2}x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, \frac{9}{2} for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-2\left(-56\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -56.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+112}}{2\times \frac{1}{2}}$$

Add \frac{81}{4} to 112.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{529}{4}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{529}{4}.

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{1}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} when ± is plus. Add -\frac{9}{2} to \frac{23}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{7}{1}$$

Divide 7 by 1.

$$x=7$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} when ± is minus. Subtract \frac{23}{2} from -\frac{9}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{16}{1}$$

Divide -16 by 1.

$$x=-16$$

The equation is now solved.

$$x=7 x=-16$$

Steps for Completing the Square

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

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

Use the distributive property to multiply x+1 by x+4 and combine like terms.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

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

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x=60$$

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

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x=60$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x=60$$

Combine 5x and -\frac{1}{2}x to get \frac{9}{2}x.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4=60$$

Subtract 4 from both sides.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=60-4$$

Subtract 4 from 60 to get 56.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=56$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+\frac{9}{2}x}{\frac{1}{2}}=\frac{56}{\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{9}{2}}{\frac{1}{2}}x=\frac{56}{\frac{1}{2}}$$

Divide \frac{9}{2} by \frac{1}{2} by multiplying \frac{9}{2} by the reciprocal of \frac{1}{2}.

$$x^{2}+9x=\frac{56}{\frac{1}{2}}$$

Divide 56 by \frac{1}{2} by multiplying 56 by the reciprocal of \frac{1}{2}.

$$x^{2}+9x=112$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=112+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=112+\frac{81}{4}$$

Add 112 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{529}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{529}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{23}{2} x+\frac{9}{2}=-\frac{23}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=7 x=-16$$