Solve (x-40)(140-10x+520)=200

$$( x - 40 ) ( 140 - 10 x + 520 ) = 200$$

Solve for x

$$x=\sqrt{149}+53\approx 65.206555616$$

$$x=53-\sqrt{149}\approx 40.793444384$$

Steps Using the Quadratic Formula

Add 140 and 520 to get 660.

$$\left(x-40\right)\left(660-10x\right)=200$$

Use the distributive property to multiply x-40 by 660-10x and combine like terms.

$$1060x-10x^{2}-26400=200$$

Subtract 200 from both sides.

$$1060x-10x^{2}-26400-200=0$$

Subtract 200 from -26400 to get -26600.

$$1060x-10x^{2}-26600=0$$

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.

$$-10x^{2}+1060x-26600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1060 for b, and -26600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1060±\sqrt{1060^{2}-4\left(-10\right)\left(-26600\right)}}{2\left(-10\right)}$$

Square 1060.

$$x=\frac{-1060±\sqrt{1123600-4\left(-10\right)\left(-26600\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1060±\sqrt{1123600+40\left(-26600\right)}}{2\left(-10\right)}$$

Multiply 40 times -26600.

$$x=\frac{-1060±\sqrt{1123600-1064000}}{2\left(-10\right)}$$

Add 1123600 to -1064000.

$$x=\frac{-1060±\sqrt{59600}}{2\left(-10\right)}$$

Take the square root of 59600.

$$x=\frac{-1060±20\sqrt{149}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1060±20\sqrt{149}}{-20}$$

Now solve the equation x=\frac{-1060±20\sqrt{149}}{-20} when ± is plus. Add -1060 to 20\sqrt{149}.

$$x=\frac{20\sqrt{149}-1060}{-20}$$

Divide -1060+20\sqrt{149} by -20.

$$x=53-\sqrt{149}$$

Now solve the equation x=\frac{-1060±20\sqrt{149}}{-20} when ± is minus. Subtract 20\sqrt{149} from -1060.

$$x=\frac{-20\sqrt{149}-1060}{-20}$$

Divide -1060-20\sqrt{149} by -20.

$$x=\sqrt{149}+53$$

The equation is now solved.

$$x=53-\sqrt{149} x=\sqrt{149}+53$$

Steps for Completing the Square

Add 140 and 520 to get 660.

$$\left(x-40\right)\left(660-10x\right)=200$$

Use the distributive property to multiply x-40 by 660-10x and combine like terms.

$$1060x-10x^{2}-26400=200$$

Add 26400 to both sides.

$$1060x-10x^{2}=200+26400$$

Add 200 and 26400 to get 26600.

$$1060x-10x^{2}=26600$$

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.

$$-10x^{2}+1060x=26600$$

Divide both sides by -10.

$$\frac{-10x^{2}+1060x}{-10}=\frac{26600}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1060}{-10}x=\frac{26600}{-10}$$

Divide 1060 by -10.

$$x^{2}-106x=\frac{26600}{-10}$$

Divide 26600 by -10.

$$x^{2}-106x=-2660$$

Divide -106, the coefficient of the x term, by 2 to get -53. Then add the square of -53 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-106x+\left(-53\right)^{2}=-2660+\left(-53\right)^{2}$$

Square -53.

$$x^{2}-106x+2809=-2660+2809$$

Add -2660 to 2809.

$$x^{2}-106x+2809=149$$

Factor x^{2}-106x+2809. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-53\right)^{2}=149$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-53\right)^{2}}=\sqrt{149}$$

Simplify.

$$x-53=\sqrt{149} x-53=-\sqrt{149}$$

Add 53 to both sides of the equation.

$$x=\sqrt{149}+53 x=53-\sqrt{149}$$