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

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

Solve for x

$$x=10$$

$$x=24$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-6 by -10x+280 and combine like terms.

$$-10x^{2}+340x-1680=720$$

Subtract 720 from both sides.

$$-10x^{2}+340x-1680-720=0$$

Subtract 720 from -1680 to get -2400.

$$-10x^{2}+340x-2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 340 for b, and -2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-340±\sqrt{340^{2}-4\left(-10\right)\left(-2400\right)}}{2\left(-10\right)}$$

Square 340.

$$x=\frac{-340±\sqrt{115600-4\left(-10\right)\left(-2400\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-340±\sqrt{115600+40\left(-2400\right)}}{2\left(-10\right)}$$

Multiply 40 times -2400.

$$x=\frac{-340±\sqrt{115600-96000}}{2\left(-10\right)}$$

Add 115600 to -96000.

$$x=\frac{-340±\sqrt{19600}}{2\left(-10\right)}$$

Take the square root of 19600.

$$x=\frac{-340±140}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-340±140}{-20}$$

Now solve the equation x=\frac{-340±140}{-20} when ± is plus. Add -340 to 140.

$$x=-\frac{200}{-20}$$

Divide -200 by -20.

$$x=10$$

Now solve the equation x=\frac{-340±140}{-20} when ± is minus. Subtract 140 from -340.

$$x=-\frac{480}{-20}$$

Divide -480 by -20.

$$x=24$$

The equation is now solved.

$$x=10 x=24$$

Steps for Completing the Square

Use the distributive property to multiply x-6 by -10x+280 and combine like terms.

$$-10x^{2}+340x-1680=720$$

Add 1680 to both sides.

$$-10x^{2}+340x=720+1680$$

Add 720 and 1680 to get 2400.

$$-10x^{2}+340x=2400$$

Divide both sides by -10.

$$\frac{-10x^{2}+340x}{-10}=\frac{2400}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{340}{-10}x=\frac{2400}{-10}$$

Divide 340 by -10.

$$x^{2}-34x=\frac{2400}{-10}$$

Divide 2400 by -10.

$$x^{2}-34x=-240$$

Divide -34, the coefficient of the x term, by 2 to get -17. Then add the square of -17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-34x+\left(-17\right)^{2}=-240+\left(-17\right)^{2}$$

Square -17.

$$x^{2}-34x+289=-240+289$$

Add -240 to 289.

$$x^{2}-34x+289=49$$

Factor x^{2}-34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-17\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-17\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-17=7 x-17=-7$$

Add 17 to both sides of the equation.

$$x=24 x=10$$