Solve 1/2(7+x)(x)=60

$$\frac { 1 } { 2 } ( 7 + x ) ( x ) = 60$$

Solve for x

$$x=-15$$

$$x=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2} by 7+x.

$$\left(\frac{1}{2}\times 7+\frac{1}{2}x\right)x=60$$

Multiply \frac{1}{2} and 7 to get \frac{7}{2}.

$$\left(\frac{7}{2}+\frac{1}{2}x\right)x=60$$

Use the distributive property to multiply \frac{7}{2}+\frac{1}{2}x by x.

$$\frac{7}{2}x+\frac{1}{2}xx=60$$

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

$$\frac{7}{2}x+\frac{1}{2}x^{2}=60$$

Subtract 60 from both sides.

$$\frac{7}{2}x+\frac{1}{2}x^{2}-60=0$$

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.

$$\frac{1}{2}x^{2}+\frac{7}{2}x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, \frac{7}{2} for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{7}{2}±\sqrt{\left(\frac{7}{2}\right)^{2}-4\times \frac{1}{2}\left(-60\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-4\times \frac{1}{2}\left(-60\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}-2\left(-60\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -60.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{49}{4}+120}}{2\times \frac{1}{2}}$$

Add \frac{49}{4} to 120.

$$x=\frac{-\frac{7}{2}±\sqrt{\frac{529}{4}}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{7}{2}±\frac{23}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{7}{2}±\frac{23}{2}}{1}$$

Now solve the equation x=\frac{-\frac{7}{2}±\frac{23}{2}}{1} when ± is plus. Add -\frac{7}{2} to \frac{23}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 8 by 1.

$$x=8$$

Now solve the equation x=\frac{-\frac{7}{2}±\frac{23}{2}}{1} when ± is minus. Subtract \frac{23}{2} from -\frac{7}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{15}{1}$$

Divide -15 by 1.

$$x=-15$$

The equation is now solved.

$$x=8 x=-15$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2} by 7+x.

$$\left(\frac{1}{2}\times 7+\frac{1}{2}x\right)x=60$$

Multiply \frac{1}{2} and 7 to get \frac{7}{2}.

$$\left(\frac{7}{2}+\frac{1}{2}x\right)x=60$$

Use the distributive property to multiply \frac{7}{2}+\frac{1}{2}x by x.

$$\frac{7}{2}x+\frac{1}{2}xx=60$$

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

$$\frac{7}{2}x+\frac{1}{2}x^{2}=60$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{1}{2}x^{2}+\frac{7}{2}x=60$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+\frac{7}{2}x}{\frac{1}{2}}=\frac{60}{\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{7}{2}}{\frac{1}{2}}x=\frac{60}{\frac{1}{2}}$$

Divide \frac{7}{2} by \frac{1}{2} by multiplying \frac{7}{2} by the reciprocal of \frac{1}{2}.

$$x^{2}+7x=\frac{60}{\frac{1}{2}}$$

Divide 60 by \frac{1}{2} by multiplying 60 by the reciprocal of \frac{1}{2}.

$$x^{2}+7x=120$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=120+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=120+\frac{49}{4}$$

Add 120 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{529}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{529}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{23}{2} x+\frac{7}{2}=-\frac{23}{2}$$

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

$$x=8 x=-15$$