Solve (20-x)x=75+5/2x

$$( 20 - x ) x = 75 + \frac { 5 } { 2 } x$$

Solve for x

$$x=10$$

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by x.

$$20x-x^{2}=75+\frac{5}{2}x$$

Subtract 75 from both sides.

$$20x-x^{2}-75=\frac{5}{2}x$$

Subtract \frac{5}{2}x from both sides.

$$20x-x^{2}-75-\frac{5}{2}x=0$$

Combine 20x and -\frac{5}{2}x to get \frac{35}{2}x.

$$\frac{35}{2}x-x^{2}-75=0$$

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.

$$-x^{2}+\frac{35}{2}x-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{35}{2} for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{35}{2}±\sqrt{\left(\frac{35}{2}\right)^{2}-4\left(-1\right)\left(-75\right)}}{2\left(-1\right)}$$

Square \frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{35}{2}±\sqrt{\frac{1225}{4}-4\left(-1\right)\left(-75\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{35}{2}±\sqrt{\frac{1225}{4}+4\left(-75\right)}}{2\left(-1\right)}$$

Multiply 4 times -75.

$$x=\frac{-\frac{35}{2}±\sqrt{\frac{1225}{4}-300}}{2\left(-1\right)}$$

Add \frac{1225}{4} to -300.

$$x=\frac{-\frac{35}{2}±\sqrt{\frac{25}{4}}}{2\left(-1\right)}$$

Take the square root of \frac{25}{4}.

$$x=\frac{-\frac{35}{2}±\frac{5}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{35}{2}±\frac{5}{2}}{-2}$$

Now solve the equation x=\frac{-\frac{35}{2}±\frac{5}{2}}{-2} when ± is plus. Add -\frac{35}{2} to \frac{5}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{15}{-2}$$

Divide -15 by -2.

$$x=\frac{15}{2}$$

Now solve the equation x=\frac{-\frac{35}{2}±\frac{5}{2}}{-2} when ± is minus. Subtract \frac{5}{2} from -\frac{35}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=\frac{15}{2} x=10$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by x.

$$20x-x^{2}=75+\frac{5}{2}x$$

Subtract \frac{5}{2}x from both sides.

$$20x-x^{2}-\frac{5}{2}x=75$$

Combine 20x and -\frac{5}{2}x to get \frac{35}{2}x.

$$\frac{35}{2}x-x^{2}=75$$

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.

$$-x^{2}+\frac{35}{2}x=75$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{35}{2}x}{-1}=\frac{75}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{35}{2}}{-1}x=\frac{75}{-1}$$

Divide \frac{35}{2} by -1.

$$x^{2}-\frac{35}{2}x=\frac{75}{-1}$$

Divide 75 by -1.

$$x^{2}-\frac{35}{2}x=-75$$

Divide -\frac{35}{2}, the coefficient of the x term, by 2 to get -\frac{35}{4}. Then add the square of -\frac{35}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{2}x+\left(-\frac{35}{4}\right)^{2}=-75+\left(-\frac{35}{4}\right)^{2}$$

Square -\frac{35}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{2}x+\frac{1225}{16}=-75+\frac{1225}{16}$$

Add -75 to \frac{1225}{16}.

$$x^{2}-\frac{35}{2}x+\frac{1225}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{35}{2}x+\frac{1225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x-\frac{35}{4}=\frac{5}{4} x-\frac{35}{4}=-\frac{5}{4}$$

Add \frac{35}{4} to both sides of the equation.

$$x=10 x=\frac{15}{2}$$