Solve 6000(1-x)*(1-x)=4860

$$6000(1-x) \times (1-x)=4860$$

Solve for x

$$x=\frac{1}{10}=0.1$$

$$x = \frac{19}{10} = 1\frac{9}{10} = 1.9$$

Steps Using the Quadratic Formula

Multiply 1-x and 1-x to get \left(1-x\right)^{2}.

$$6000\left(1-x\right)^{2}=4860$$

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

$$6000\left(1-2x+x^{2}\right)=4860$$

Use the distributive property to multiply 6000 by 1-2x+x^{2}.

$$6000-12000x+6000x^{2}=4860$$

Subtract 4860 from both sides.

$$6000-12000x+6000x^{2}-4860=0$$

Subtract 4860 from 6000 to get 1140.

$$1140-12000x+6000x^{2}=0$$

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.

$$6000x^{2}-12000x+1140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6000 for a, -12000 for b, and 1140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12000\right)±\sqrt{\left(-12000\right)^{2}-4\times 6000\times 1140}}{2\times 6000}$$

Square -12000.

$$x=\frac{-\left(-12000\right)±\sqrt{144000000-4\times 6000\times 1140}}{2\times 6000}$$

Multiply -4 times 6000.

$$x=\frac{-\left(-12000\right)±\sqrt{144000000-24000\times 1140}}{2\times 6000}$$

Multiply -24000 times 1140.

$$x=\frac{-\left(-12000\right)±\sqrt{144000000-27360000}}{2\times 6000}$$

Add 144000000 to -27360000.

$$x=\frac{-\left(-12000\right)±\sqrt{116640000}}{2\times 6000}$$

Take the square root of 116640000.

$$x=\frac{-\left(-12000\right)±10800}{2\times 6000}$$

The opposite of -12000 is 12000.

$$x=\frac{12000±10800}{2\times 6000}$$

Multiply 2 times 6000.

$$x=\frac{12000±10800}{12000}$$

Now solve the equation x=\frac{12000±10800}{12000} when ± is plus. Add 12000 to 10800.

$$x=\frac{22800}{12000}$$

Reduce the fraction \frac{22800}{12000} to lowest terms by extracting and canceling out 1200.

$$x=\frac{19}{10}$$

Now solve the equation x=\frac{12000±10800}{12000} when ± is minus. Subtract 10800 from 12000.

$$x=\frac{1200}{12000}$$

Reduce the fraction \frac{1200}{12000} to lowest terms by extracting and canceling out 1200.

$$x=\frac{1}{10}$$

The equation is now solved.

$$x=\frac{19}{10} x=\frac{1}{10}$$

Steps for Completing the Square

Multiply 1-x and 1-x to get \left(1-x\right)^{2}.

$$6000\left(1-x\right)^{2}=4860$$

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

$$6000\left(1-2x+x^{2}\right)=4860$$

Use the distributive property to multiply 6000 by 1-2x+x^{2}.

$$6000-12000x+6000x^{2}=4860$$

Subtract 6000 from both sides.

$$-12000x+6000x^{2}=4860-6000$$

Subtract 6000 from 4860 to get -1140.

$$-12000x+6000x^{2}=-1140$$

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.

$$6000x^{2}-12000x=-1140$$

Divide both sides by 6000.

$$\frac{6000x^{2}-12000x}{6000}=-\frac{1140}{6000}$$

Dividing by 6000 undoes the multiplication by 6000.

$$x^{2}+\left(-\frac{12000}{6000}\right)x=-\frac{1140}{6000}$$

Divide -12000 by 6000.

$$x^{2}-2x=-\frac{1140}{6000}$$

Reduce the fraction \frac{-1140}{6000} to lowest terms by extracting and canceling out 60.

$$x^{2}-2x=-\frac{19}{100}$$

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.

$$x^{2}-2x+1=-\frac{19}{100}+1$$

Add -\frac{19}{100} to 1.

$$x^{2}-2x+1=\frac{81}{100}$$

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}.

$$\left(x-1\right)^{2}=\frac{81}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{81}{100}}$$

Simplify.

$$x-1=\frac{9}{10} x-1=-\frac{9}{10}$$

Add 1 to both sides of the equation.

$$x=\frac{19}{10} x=\frac{1}{10}$$