Solve (280-10x)x-6x=720

$$( 280 - 10 x ) x - 6 x = 720$$

Solve for x

$$x = \frac{\sqrt{11569} + 137}{10} \approx 24.455928598$$

$$x = \frac{137 - \sqrt{11569}}{10} \approx 2.944071402$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 280-10x by x.

$$280x-10x^{2}-6x=720$$

Combine 280x and -6x to get 274x.

$$274x-10x^{2}=720$$

Subtract 720 from both sides.

$$274x-10x^{2}-720=0$$

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.

$$-10x^{2}+274x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 274 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-274±\sqrt{274^{2}-4\left(-10\right)\left(-720\right)}}{2\left(-10\right)}$$

Square 274.

$$x=\frac{-274±\sqrt{75076-4\left(-10\right)\left(-720\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-274±\sqrt{75076+40\left(-720\right)}}{2\left(-10\right)}$$

Multiply 40 times -720.

$$x=\frac{-274±\sqrt{75076-28800}}{2\left(-10\right)}$$

Add 75076 to -28800.

$$x=\frac{-274±\sqrt{46276}}{2\left(-10\right)}$$

Take the square root of 46276.

$$x=\frac{-274±2\sqrt{11569}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-274±2\sqrt{11569}}{-20}$$

Now solve the equation x=\frac{-274±2\sqrt{11569}}{-20} when ± is plus. Add -274 to 2\sqrt{11569}.

$$x=\frac{2\sqrt{11569}-274}{-20}$$

Divide -274+2\sqrt{11569} by -20.

$$x=\frac{137-\sqrt{11569}}{10}$$

Now solve the equation x=\frac{-274±2\sqrt{11569}}{-20} when ± is minus. Subtract 2\sqrt{11569} from -274.

$$x=\frac{-2\sqrt{11569}-274}{-20}$$

Divide -274-2\sqrt{11569} by -20.

$$x=\frac{\sqrt{11569}+137}{10}$$

The equation is now solved.

$$x=\frac{137-\sqrt{11569}}{10} x=\frac{\sqrt{11569}+137}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 280-10x by x.

$$280x-10x^{2}-6x=720$$

Combine 280x and -6x to get 274x.

$$274x-10x^{2}=720$$

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.

$$-10x^{2}+274x=720$$

Divide both sides by -10.

$$\frac{-10x^{2}+274x}{-10}=\frac{720}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{274}{-10}x=\frac{720}{-10}$$

Reduce the fraction \frac{274}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{137}{5}x=\frac{720}{-10}$$

Divide 720 by -10.

$$x^{2}-\frac{137}{5}x=-72$$

Divide -\frac{137}{5}, the coefficient of the x term, by 2 to get -\frac{137}{10}. Then add the square of -\frac{137}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{137}{5}x+\left(-\frac{137}{10}\right)^{2}=-72+\left(-\frac{137}{10}\right)^{2}$$

Square -\frac{137}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{137}{5}x+\frac{18769}{100}=-72+\frac{18769}{100}$$

Add -72 to \frac{18769}{100}.

$$x^{2}-\frac{137}{5}x+\frac{18769}{100}=\frac{11569}{100}$$

Factor x^{2}-\frac{137}{5}x+\frac{18769}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{137}{10}\right)^{2}=\frac{11569}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{137}{10}\right)^{2}}=\sqrt{\frac{11569}{100}}$$

Simplify.

$$x-\frac{137}{10}=\frac{\sqrt{11569}}{10} x-\frac{137}{10}=-\frac{\sqrt{11569}}{10}$$

Add \frac{137}{10} to both sides of the equation.

$$x=\frac{\sqrt{11569}+137}{10} x=\frac{137-\sqrt{11569}}{10}$$