Solve (20-5x)(6-x)=12.5*9

$$(20-5x)(6-x)=12.5 \times 9$$

Solve for x

$$x=\frac{\sqrt{94}}{2}+5\approx 9.847679857$$

$$x=-\frac{\sqrt{94}}{2}+5\approx 0.152320143$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-5x by 6-x and combine like terms.

$$120-50x+5x^{2}=12.5\times 9$$

Multiply 12.5 and 9 to get 112.5.

$$120-50x+5x^{2}=112.5$$

Subtract 112.5 from both sides.

$$120-50x+5x^{2}-112.5=0$$

Subtract 112.5 from 120 to get 7.5.

$$7.5-50x+5x^{2}=0$$

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.

$$5x^{2}-50x+7.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -50 for b, and 7.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 5\times 7.5}}{2\times 5}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 5\times 7.5}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-50\right)±\sqrt{2500-20\times 7.5}}{2\times 5}$$

Multiply -20 times 7.5.

$$x=\frac{-\left(-50\right)±\sqrt{2500-150}}{2\times 5}$$

Add 2500 to -150.

$$x=\frac{-\left(-50\right)±\sqrt{2350}}{2\times 5}$$

Take the square root of 2350.

$$x=\frac{-\left(-50\right)±5\sqrt{94}}{2\times 5}$$

The opposite of -50 is 50.

$$x=\frac{50±5\sqrt{94}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{50±5\sqrt{94}}{10}$$

Now solve the equation x=\frac{50±5\sqrt{94}}{10} when ± is plus. Add 50 to 5\sqrt{94}.

$$x=\frac{5\sqrt{94}+50}{10}$$

Divide 50+5\sqrt{94} by 10.

$$x=\frac{\sqrt{94}}{2}+5$$

Now solve the equation x=\frac{50±5\sqrt{94}}{10} when ± is minus. Subtract 5\sqrt{94} from 50.

$$x=\frac{50-5\sqrt{94}}{10}$$

Divide 50-5\sqrt{94} by 10.

$$x=-\frac{\sqrt{94}}{2}+5$$

The equation is now solved.

$$x=\frac{\sqrt{94}}{2}+5 x=-\frac{\sqrt{94}}{2}+5$$

Steps for Completing the Square

Use the distributive property to multiply 20-5x by 6-x and combine like terms.

$$120-50x+5x^{2}=12.5\times 9$$

Multiply 12.5 and 9 to get 112.5.

$$120-50x+5x^{2}=112.5$$

Subtract 120 from both sides.

$$-50x+5x^{2}=112.5-120$$

Subtract 120 from 112.5 to get -7.5.

$$-50x+5x^{2}=-7.5$$

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.

$$5x^{2}-50x=-7.5$$

Divide both sides by 5.

$$\frac{5x^{2}-50x}{5}=-\frac{7.5}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{50}{5}\right)x=-\frac{7.5}{5}$$

Divide -50 by 5.

$$x^{2}-10x=-\frac{7.5}{5}$$

Divide -7.5 by 5.

$$x^{2}-10x=-1.5$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-1.5+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-1.5+25$$

Add -1.5 to 25.

$$x^{2}-10x+25=23.5$$

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}.

$$\left(x-5\right)^{2}=23.5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{23.5}$$

Simplify.

$$x-5=\frac{\sqrt{94}}{2} x-5=-\frac{\sqrt{94}}{2}$$

Add 5 to both sides of the equation.

$$x=\frac{\sqrt{94}}{2}+5 x=-\frac{\sqrt{94}}{2}+5$$