Solve (100+10x)(60-x)=2240

$$( 100 + 10 x ) ( 60 - x ) = 2240$$

Solve for x

$$x=\sqrt{1001}+25\approx 56.638584039$$

$$x=25-\sqrt{1001}\approx -6.638584039$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100+10x by 60-x and combine like terms.

$$6000+500x-10x^{2}=2240$$

Subtract 2240 from both sides.

$$6000+500x-10x^{2}-2240=0$$

Subtract 2240 from 6000 to get 3760.

$$3760+500x-10x^{2}=0$$

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.

$$-10x^{2}+500x+3760=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 500 for b, and 3760 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-500±\sqrt{500^{2}-4\left(-10\right)\times 3760}}{2\left(-10\right)}$$

Square 500.

$$x=\frac{-500±\sqrt{250000-4\left(-10\right)\times 3760}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-500±\sqrt{250000+40\times 3760}}{2\left(-10\right)}$$

Multiply 40 times 3760.

$$x=\frac{-500±\sqrt{250000+150400}}{2\left(-10\right)}$$

Add 250000 to 150400.

$$x=\frac{-500±\sqrt{400400}}{2\left(-10\right)}$$

Take the square root of 400400.

$$x=\frac{-500±20\sqrt{1001}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-500±20\sqrt{1001}}{-20}$$

Now solve the equation x=\frac{-500±20\sqrt{1001}}{-20} when ± is plus. Add -500 to 20\sqrt{1001}.

$$x=\frac{20\sqrt{1001}-500}{-20}$$

Divide -500+20\sqrt{1001} by -20.

$$x=25-\sqrt{1001}$$

Now solve the equation x=\frac{-500±20\sqrt{1001}}{-20} when ± is minus. Subtract 20\sqrt{1001} from -500.

$$x=\frac{-20\sqrt{1001}-500}{-20}$$

Divide -500-20\sqrt{1001} by -20.

$$x=\sqrt{1001}+25$$

The equation is now solved.

$$x=25-\sqrt{1001} x=\sqrt{1001}+25$$

Steps for Completing the Square

Use the distributive property to multiply 100+10x by 60-x and combine like terms.

$$6000+500x-10x^{2}=2240$$

Subtract 6000 from both sides.

$$500x-10x^{2}=2240-6000$$

Subtract 6000 from 2240 to get -3760.

$$500x-10x^{2}=-3760$$

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.

$$-10x^{2}+500x=-3760$$

Divide both sides by -10.

$$\frac{-10x^{2}+500x}{-10}=-\frac{3760}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{500}{-10}x=-\frac{3760}{-10}$$

Divide 500 by -10.

$$x^{2}-50x=-\frac{3760}{-10}$$

Divide -3760 by -10.

$$x^{2}-50x=376$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=376+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=376+625$$

Add 376 to 625.

$$x^{2}-50x+625=1001$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=1001$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{1001}$$

Simplify.

$$x-25=\sqrt{1001} x-25=-\sqrt{1001}$$

Add 25 to both sides of the equation.

$$x=\sqrt{1001}+25 x=25-\sqrt{1001}$$