Solve 1/2(4-x)(6-x)=1

$$\frac{ 1 }{ 2 } (4-x)(6-x)=1$$

Solve for x

$$x=5-\sqrt{3}\approx 3.267949192$$

$$x=\sqrt{3}+5\approx 6.732050808$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2} by 4-x.

$$\left(2-\frac{1}{2}x\right)\left(6-x\right)=1$$

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

$$12-5x+\frac{1}{2}x^{2}=1$$

Subtract 1 from both sides.

$$12-5x+\frac{1}{2}x^{2}-1=0$$

Subtract 1 from 12 to get 11.

$$11-5x+\frac{1}{2}x^{2}=0$$

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.

$$\frac{1}{2}x^{2}-5x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -5 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times \frac{1}{2}\times 11}}{2\times \frac{1}{2}}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times \frac{1}{2}\times 11}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-5\right)±\sqrt{25-2\times 11}}{2\times \frac{1}{2}}$$

Multiply -2 times 11.

$$x=\frac{-\left(-5\right)±\sqrt{25-22}}{2\times \frac{1}{2}}$$

Add 25 to -22.

$$x=\frac{-\left(-5\right)±\sqrt{3}}{2\times \frac{1}{2}}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{3}}{2\times \frac{1}{2}}$$

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

$$x=\frac{5±\sqrt{3}}{1}$$

Now solve the equation x=\frac{5±\sqrt{3}}{1} when ± is plus. Add 5 to \sqrt{3}.

$$x=\frac{\sqrt{3}+5}{1}$$

Divide 5+\sqrt{3} by 1.

$$x=\sqrt{3}+5$$

Now solve the equation x=\frac{5±\sqrt{3}}{1} when ± is minus. Subtract \sqrt{3} from 5.

$$x=\frac{5-\sqrt{3}}{1}$$

Divide 5-\sqrt{3} by 1.

$$x=5-\sqrt{3}$$

The equation is now solved.

$$x=\sqrt{3}+5 x=5-\sqrt{3}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2} by 4-x.

$$\left(2-\frac{1}{2}x\right)\left(6-x\right)=1$$

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

$$12-5x+\frac{1}{2}x^{2}=1$$

Subtract 12 from both sides.

$$-5x+\frac{1}{2}x^{2}=1-12$$

Subtract 12 from 1 to get -11.

$$-5x+\frac{1}{2}x^{2}=-11$$

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.

$$\frac{1}{2}x^{2}-5x=-11$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-5x}{\frac{1}{2}}=-\frac{11}{\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{5}{\frac{1}{2}}\right)x=-\frac{11}{\frac{1}{2}}$$

Divide -5 by \frac{1}{2} by multiplying -5 by the reciprocal of \frac{1}{2}.

$$x^{2}-10x=-\frac{11}{\frac{1}{2}}$$

Divide -11 by \frac{1}{2} by multiplying -11 by the reciprocal of \frac{1}{2}.

$$x^{2}-10x=-22$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-22+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-22+25$$

Add -22 to 25.

$$x^{2}-10x+25=3$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{3}$$

Simplify.

$$x-5=\sqrt{3} x-5=-\sqrt{3}$$

Add 5 to both sides of the equation.

$$x=\sqrt{3}+5 x=5-\sqrt{3}$$