Solve 1/2(80-x)*x=750

$$\frac { 1 } { 2 } ( 80 - x ) \cdot x = 750$$

Solve for x

$$x=30$$

$$x=50$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2} by 80-x.

$$\left(\frac{1}{2}\times 80+\frac{1}{2}\left(-1\right)x\right)x=750$$

Multiply \frac{1}{2} and 80 to get \frac{80}{2}.

$$\left(\frac{80}{2}+\frac{1}{2}\left(-1\right)x\right)x=750$$

Divide 80 by 2 to get 40.

$$\left(40+\frac{1}{2}\left(-1\right)x\right)x=750$$

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

$$\left(40-\frac{1}{2}x\right)x=750$$

Use the distributive property to multiply 40-\frac{1}{2}x by x.

$$40x-\frac{1}{2}xx=750$$

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

$$40x-\frac{1}{2}x^{2}=750$$

Subtract 750 from both sides.

$$40x-\frac{1}{2}x^{2}-750=0$$

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.

$$-\frac{1}{2}x^{2}+40x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, 40 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-\frac{1}{2}\right)\left(-750\right)}}{2\left(-\frac{1}{2}\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-\frac{1}{2}\right)\left(-750\right)}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-40±\sqrt{1600+2\left(-750\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -750.

$$x=\frac{-40±\sqrt{1600-1500}}{2\left(-\frac{1}{2}\right)}$$

Add 1600 to -1500.

$$x=\frac{-40±\sqrt{100}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of 100.

$$x=\frac{-40±10}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-40±10}{-1}$$

Now solve the equation x=\frac{-40±10}{-1} when ± is plus. Add -40 to 10.

$$x=-\frac{30}{-1}$$

Divide -30 by -1.

$$x=30$$

Now solve the equation x=\frac{-40±10}{-1} when ± is minus. Subtract 10 from -40.

$$x=-\frac{50}{-1}$$

Divide -50 by -1.

$$x=50$$

The equation is now solved.

$$x=30 x=50$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2} by 80-x.

$$\left(\frac{1}{2}\times 80+\frac{1}{2}\left(-1\right)x\right)x=750$$

Multiply \frac{1}{2} and 80 to get \frac{80}{2}.

$$\left(\frac{80}{2}+\frac{1}{2}\left(-1\right)x\right)x=750$$

Divide 80 by 2 to get 40.

$$\left(40+\frac{1}{2}\left(-1\right)x\right)x=750$$

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

$$\left(40-\frac{1}{2}x\right)x=750$$

Use the distributive property to multiply 40-\frac{1}{2}x by x.

$$40x-\frac{1}{2}xx=750$$

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

$$40x-\frac{1}{2}x^{2}=750$$

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.

$$-\frac{1}{2}x^{2}+40x=750$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}+40x}{-\frac{1}{2}}=\frac{750}{-\frac{1}{2}}$$

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

$$x^{2}+\frac{40}{-\frac{1}{2}}x=\frac{750}{-\frac{1}{2}}$$

Divide 40 by -\frac{1}{2} by multiplying 40 by the reciprocal of -\frac{1}{2}.

$$x^{2}-80x=\frac{750}{-\frac{1}{2}}$$

Divide 750 by -\frac{1}{2} by multiplying 750 by the reciprocal of -\frac{1}{2}.

$$x^{2}-80x=-1500$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-80x+\left(-40\right)^{2}=-1500+\left(-40\right)^{2}$$

Square -40.

$$x^{2}-80x+1600=-1500+1600$$

Add -1500 to 1600.

$$x^{2}-80x+1600=100$$

Factor x^{2}-80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-40\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-40=10 x-40=-10$$

Add 40 to both sides of the equation.

$$x=50 x=30$$