Solve 240(1+1.5x)(1+x)=800

$$240 ( 1 + 1.5 x ) ( 1 + x ) = 800$$

Solve for x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 240 by 1+1.5x.

$$\left(240+360x\right)\left(1+x\right)=800$$

Use the distributive property to multiply 240+360x by 1+x and combine like terms.

$$240+600x+360x^{2}=800$$

Subtract 800 from both sides.

$$240+600x+360x^{2}-800=0$$

Subtract 800 from 240 to get -560.

$$-560+600x+360x^{2}=0$$

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.

$$360x^{2}+600x-560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 360 for a, 600 for b, and -560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\times 360\left(-560\right)}}{2\times 360}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\times 360\left(-560\right)}}{2\times 360}$$

Multiply -4 times 360.

$$x=\frac{-600±\sqrt{360000-1440\left(-560\right)}}{2\times 360}$$

Multiply -1440 times -560.

$$x=\frac{-600±\sqrt{360000+806400}}{2\times 360}$$

Add 360000 to 806400.

$$x=\frac{-600±\sqrt{1166400}}{2\times 360}$$

Take the square root of 1166400.

$$x=\frac{-600±1080}{2\times 360}$$

Multiply 2 times 360.

$$x=\frac{-600±1080}{720}$$

Now solve the equation x=\frac{-600±1080}{720} when ± is plus. Add -600 to 1080.

$$x=\frac{480}{720}$$

Reduce the fraction \frac{480}{720} to lowest terms by extracting and canceling out 240.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{-600±1080}{720} when ± is minus. Subtract 1080 from -600.

$$x=-\frac{1680}{720}$$

Reduce the fraction \frac{-1680}{720} to lowest terms by extracting and canceling out 240.

$$x=-\frac{7}{3}$$

The equation is now solved.

$$x=\frac{2}{3} x=-\frac{7}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 240 by 1+1.5x.

$$\left(240+360x\right)\left(1+x\right)=800$$

Use the distributive property to multiply 240+360x by 1+x and combine like terms.

$$240+600x+360x^{2}=800$$

Subtract 240 from both sides.

$$600x+360x^{2}=800-240$$

Subtract 240 from 800 to get 560.

$$600x+360x^{2}=560$$

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.

$$360x^{2}+600x=560$$

Divide both sides by 360.

$$\frac{360x^{2}+600x}{360}=\frac{560}{360}$$

Dividing by 360 undoes the multiplication by 360.

$$x^{2}+\frac{600}{360}x=\frac{560}{360}$$

Reduce the fraction \frac{600}{360} to lowest terms by extracting and canceling out 120.

$$x^{2}+\frac{5}{3}x=\frac{560}{360}$$

Reduce the fraction \frac{560}{360} to lowest terms by extracting and canceling out 40.

$$x^{2}+\frac{5}{3}x=\frac{14}{9}$$

Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{14}{9}+\left(\frac{5}{6}\right)^{2}$$

Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{14}{9}+\frac{25}{36}$$

Add \frac{14}{9} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{9}{4}$$

Factor x^{2}+\frac{5}{3}x+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{5}{6}=\frac{3}{2} x+\frac{5}{6}=-\frac{3}{2}$$

Subtract \frac{5}{6} from both sides of the equation.

$$x=\frac{2}{3} x=-\frac{7}{3}$$