Solve (280-10x)x=720

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

Solve for x

$$x=2\sqrt{31}+14\approx 25.135528726$$

$$x=14-2\sqrt{31}\approx 2.864471274$$

Steps Using the Quadratic Formula

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

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

Subtract 720 from both sides.

$$280x-10x^{2}-720=0$$

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.

$$-10x^{2}+280x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 280 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-280±\sqrt{280^{2}-4\left(-10\right)\left(-720\right)}}{2\left(-10\right)}$$

Square 280.

$$x=\frac{-280±\sqrt{78400-4\left(-10\right)\left(-720\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-280±\sqrt{78400+40\left(-720\right)}}{2\left(-10\right)}$$

Multiply 40 times -720.

$$x=\frac{-280±\sqrt{78400-28800}}{2\left(-10\right)}$$

Add 78400 to -28800.

$$x=\frac{-280±\sqrt{49600}}{2\left(-10\right)}$$

Take the square root of 49600.

$$x=\frac{-280±40\sqrt{31}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-280±40\sqrt{31}}{-20}$$

Now solve the equation x=\frac{-280±40\sqrt{31}}{-20} when ± is plus. Add -280 to 40\sqrt{31}.

$$x=\frac{40\sqrt{31}-280}{-20}$$

Divide -280+40\sqrt{31} by -20.

$$x=14-2\sqrt{31}$$

Now solve the equation x=\frac{-280±40\sqrt{31}}{-20} when ± is minus. Subtract 40\sqrt{31} from -280.

$$x=\frac{-40\sqrt{31}-280}{-20}$$

Divide -280-40\sqrt{31} by -20.

$$x=2\sqrt{31}+14$$

The equation is now solved.

$$x=14-2\sqrt{31} x=2\sqrt{31}+14$$

Steps for Completing the Square

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

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

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.

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

Divide both sides by -10.

$$\frac{-10x^{2}+280x}{-10}=\frac{720}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{280}{-10}x=\frac{720}{-10}$$

Divide 280 by -10.

$$x^{2}-28x=\frac{720}{-10}$$

Divide 720 by -10.

$$x^{2}-28x=-72$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-28x+\left(-14\right)^{2}=-72+\left(-14\right)^{2}$$

Square -14.

$$x^{2}-28x+196=-72+196$$

Add -72 to 196.

$$x^{2}-28x+196=124$$

Factor x^{2}-28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-14\right)^{2}=124$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{124}$$

Simplify.

$$x-14=2\sqrt{31} x-14=-2\sqrt{31}$$

Add 14 to both sides of the equation.

$$x=2\sqrt{31}+14 x=14-2\sqrt{31}$$