Solve (10+x)(730-x)=1000

$$( 10 + x ) ( 730 - x ) = 1000$$

Solve for x

$$x=30\sqrt{151}+360\approx 728.646171823$$

$$x=360-30\sqrt{151}\approx -8.646171823$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 730-x and combine like terms.

$$7300+720x-x^{2}=1000$$

Subtract 1000 from both sides.

$$7300+720x-x^{2}-1000=0$$

Subtract 1000 from 7300 to get 6300.

$$6300+720x-x^{2}=0$$

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.

$$-x^{2}+720x+6300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 720 for b, and 6300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-720±\sqrt{720^{2}-4\left(-1\right)\times 6300}}{2\left(-1\right)}$$

Square 720.

$$x=\frac{-720±\sqrt{518400-4\left(-1\right)\times 6300}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-720±\sqrt{518400+4\times 6300}}{2\left(-1\right)}$$

Multiply 4 times 6300.

$$x=\frac{-720±\sqrt{518400+25200}}{2\left(-1\right)}$$

Add 518400 to 25200.

$$x=\frac{-720±\sqrt{543600}}{2\left(-1\right)}$$

Take the square root of 543600.

$$x=\frac{-720±60\sqrt{151}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-720±60\sqrt{151}}{-2}$$

Now solve the equation x=\frac{-720±60\sqrt{151}}{-2} when ± is plus. Add -720 to 60\sqrt{151}.

$$x=\frac{60\sqrt{151}-720}{-2}$$

Divide -720+60\sqrt{151} by -2.

$$x=360-30\sqrt{151}$$

Now solve the equation x=\frac{-720±60\sqrt{151}}{-2} when ± is minus. Subtract 60\sqrt{151} from -720.

$$x=\frac{-60\sqrt{151}-720}{-2}$$

Divide -720-60\sqrt{151} by -2.

$$x=30\sqrt{151}+360$$

The equation is now solved.

$$x=360-30\sqrt{151} x=30\sqrt{151}+360$$

Steps for Completing the Square

Use the distributive property to multiply 10+x by 730-x and combine like terms.

$$7300+720x-x^{2}=1000$$

Subtract 7300 from both sides.

$$720x-x^{2}=1000-7300$$

Subtract 7300 from 1000 to get -6300.

$$720x-x^{2}=-6300$$

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.

$$-x^{2}+720x=-6300$$

Divide both sides by -1.

$$\frac{-x^{2}+720x}{-1}=-\frac{6300}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{720}{-1}x=-\frac{6300}{-1}$$

Divide 720 by -1.

$$x^{2}-720x=-\frac{6300}{-1}$$

Divide -6300 by -1.

$$x^{2}-720x=6300$$

Divide -720, the coefficient of the x term, by 2 to get -360. Then add the square of -360 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-720x+\left(-360\right)^{2}=6300+\left(-360\right)^{2}$$

Square -360.

$$x^{2}-720x+129600=6300+129600$$

Add 6300 to 129600.

$$x^{2}-720x+129600=135900$$

Factor x^{2}-720x+129600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-360\right)^{2}=135900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-360\right)^{2}}=\sqrt{135900}$$

Simplify.

$$x-360=30\sqrt{151} x-360=-30\sqrt{151}$$

Add 360 to both sides of the equation.

$$x=30\sqrt{151}+360 x=360-30\sqrt{151}$$