Solve 1/2*(6-x)(8-x)=12

$$\frac { 1 } { 2 } \times ( 6 - x ) ( 8 - x ) = 12$$

Solve for x

$$x=12$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2} by 6-x.

$$\left(3-\frac{1}{2}x\right)\left(8-x\right)=12$$

Use the distributive property to multiply 3-\frac{1}{2}x by 8-x and combine like terms.

$$24-7x+\frac{1}{2}x^{2}=12$$

Subtract 12 from both sides.

$$24-7x+\frac{1}{2}x^{2}-12=0$$

Subtract 12 from 24 to get 12.

$$12-7x+\frac{1}{2}x^{2}=0$$

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.

$$\frac{1}{2}x^{2}-7x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times \frac{1}{2}\times 12}}{2\times \frac{1}{2}}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times \frac{1}{2}\times 12}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-7\right)±\sqrt{49-2\times 12}}{2\times \frac{1}{2}}$$

Multiply -2 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2\times \frac{1}{2}}$$

Add 49 to -24.

$$x=\frac{-\left(-7\right)±\sqrt{25}}{2\times \frac{1}{2}}$$

Take the square root of 25.

$$x=\frac{-\left(-7\right)±5}{2\times \frac{1}{2}}$$

The opposite of -7 is 7.

$$x=\frac{7±5}{2\times \frac{1}{2}}$$

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

$$x=\frac{7±5}{1}$$

Now solve the equation x=\frac{7±5}{1} when ± is plus. Add 7 to 5.

$$x=\frac{12}{1}$$

Divide 12 by 1.

$$x=12$$

Now solve the equation x=\frac{7±5}{1} when ± is minus. Subtract 5 from 7.

$$x=\frac{2}{1}$$

Divide 2 by 1.

$$x=2$$

The equation is now solved.

$$x=12 x=2$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2} by 6-x.

$$\left(3-\frac{1}{2}x\right)\left(8-x\right)=12$$

Use the distributive property to multiply 3-\frac{1}{2}x by 8-x and combine like terms.

$$24-7x+\frac{1}{2}x^{2}=12$$

Subtract 24 from both sides.

$$-7x+\frac{1}{2}x^{2}=12-24$$

Subtract 24 from 12 to get -12.

$$-7x+\frac{1}{2}x^{2}=-12$$

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.

$$\frac{1}{2}x^{2}-7x=-12$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-7x}{\frac{1}{2}}=-\frac{12}{\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{7}{\frac{1}{2}}\right)x=-\frac{12}{\frac{1}{2}}$$

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

$$x^{2}-14x=-\frac{12}{\frac{1}{2}}$$

Divide -12 by \frac{1}{2} by multiplying -12 by the reciprocal of \frac{1}{2}.

$$x^{2}-14x=-24$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-24+49$$

Add -24 to 49.

$$x^{2}-14x+49=25$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-7=5 x-7=-5$$

Add 7 to both sides of the equation.

$$x=12 x=2$$