Solve 7500^2left(1+xright)^2=10800

$${ 7500 }^{ 2 } { \left(1+x \right) }^{ 2 } =10800$$

Solve for x

$$x=\frac{\sqrt{3}}{125}-1\approx -0.986143594$$

$$x=-\frac{\sqrt{3}}{125}-1\approx -1.013856406$$

Steps Using the Quadratic Formula

Calculate 7500 to the power of 2 and get 56250000.

$$56250000\left(1+x\right)^{2}=10800$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$56250000\left(1+2x+x^{2}\right)=10800$$

Use the distributive property to multiply 56250000 by 1+2x+x^{2}.

$$56250000+112500000x+56250000x^{2}=10800$$

Subtract 10800 from both sides.

$$56250000+112500000x+56250000x^{2}-10800=0$$

Subtract 10800 from 56250000 to get 56239200.

$$56239200+112500000x+56250000x^{2}=0$$

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.

$$56250000x^{2}[phone]x+56239200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 56250000 for a, 112500000 for b, and 56239200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112500000±\sqrt{112500000^{2}-4\times 56250000\times 56239200}}{2\times 56250000}$$

Square 112500000.

$$x=\frac{-112500000±\sqrt{12656250000000000-4\times 56250000\times 56239200}}{2\times 56250000}$$

Multiply -4 times 56250000.

$$x=\frac{-112500000±\sqrt{12656250000000000-225000000\times 56239200}}{2\times 56250000}$$

Multiply -225000000 times 56239200.

$$x=\frac{-112500000±\sqrt{12656250000000000-12653820000000000}}{2\times 56250000}$$

Add 12656250000000000 to -12653820000000000.

$$x=\frac{-112500000±\sqrt{2430000000000}}{2\times 56250000}$$

Take the square root of 2430000000000.

$$x=\frac{-112500000±900000\sqrt{3}}{2\times 56250000}$$

Multiply 2 times 56250000.

$$x=\frac{-112500000±900000\sqrt{3}}{112500000}$$

Now solve the equation x=\frac{-112500000±900000\sqrt{3}}{112500000} when ± is plus. Add -112500000 to 900000\sqrt{3}.

$$x=\frac{900000\sqrt{3}-112500000}{112500000}$$

Divide -112500000+900000\sqrt{3} by 112500000.

$$x=\frac{\sqrt{3}}{125}-1$$

Now solve the equation x=\frac{-112500000±900000\sqrt{3}}{112500000} when ± is minus. Subtract 900000\sqrt{3} from -112500000.

$$x=\frac{-900000\sqrt{3}-112500000}{112500000}$$

Divide -112500000-900000\sqrt{3} by 112500000.

$$x=-\frac{\sqrt{3}}{125}-1$$

The equation is now solved.

$$x=\frac{\sqrt{3}}{125}-1 x=-\frac{\sqrt{3}}{125}-1$$

Steps for Completing the Square

Calculate 7500 to the power of 2 and get 56250000.

$$56250000\left(1+x\right)^{2}=10800$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$56250000\left(1+2x+x^{2}\right)=10800$$

Use the distributive property to multiply 56250000 by 1+2x+x^{2}.

$$56250000+112500000x+56250000x^{2}=10800$$

Subtract 56250000 from both sides.

$$112500000x+56250000x^{2}=10800-56250000$$

Subtract 56250000 from 10800 to get -56239200.

$$112500000x+56250000x^{2}=-56239200$$

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.

$$56250000x^{2}[phone]x=-56239200$$

Divide both sides by 56250000.

$$\frac{56250000x^{2}[phone]x}{56250000}=-\frac{56239200}{56250000}$$

Dividing by 56250000 undoes the multiplication by 56250000.

$$x^{2}+\frac{112500000}{56250000}x=-\frac{56239200}{56250000}$$

Divide 112500000 by 56250000.

$$x^{2}+2x=-\frac{56239200}{56250000}$$

Reduce the fraction \frac{-56239200}{56250000} to lowest terms by extracting and canceling out 3600.

$$x^{2}+2x=-\frac{15622}{15625}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=-\frac{15622}{15625}+1^{2}$$

Square 1.

$$x^{2}+2x+1=-\frac{15622}{15625}+1$$

Add -\frac{15622}{15625} to 1.

$$x^{2}+2x+1=\frac{3}{15625}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=\frac{3}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{3}{15625}}$$

Simplify.

$$x+1=\frac{\sqrt{3}}{125} x+1=-\frac{\sqrt{3}}{125}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{3}}{125}-1 x=-\frac{\sqrt{3}}{125}-1$$