Solve 112=100x*(1+0.00001*(20-x))

$$112=100x \times (1+0.00001 \times (20-x))$$

Solve for x

$$x=10\sqrt{25008881}+50010\approx 100018.880211418$$

$$x=50010-10\sqrt{25008881}\approx 1.119788582$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.00001 by 20-x.

$$112=100x\left(1+0.0002-0.00001x\right)$$

Add 1 and 0.0002 to get 1.0002.

$$112=100x\left(1.0002-0.00001x\right)$$

Use the distributive property to multiply 100x by 1.0002-0.00001x.

$$112=100.02x-0.001x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$100.02x-0.001x^{2}=112$$

Subtract 112 from both sides.

$$100.02x-0.001x^{2}-112=0$$

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.

$$-0.001x^{2}+100.02x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.001 for a, 100.02 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100.02±\sqrt{100.02^{2}-4\left(-0.001\right)\left(-112\right)}}{2\left(-0.001\right)}$$

Square 100.02 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-100.02±\sqrt{10004.0004-4\left(-0.001\right)\left(-112\right)}}{2\left(-0.001\right)}$$

Multiply -4 times -0.001.

$$x=\frac{-100.02±\sqrt{10004.0004+0.004\left(-112\right)}}{2\left(-0.001\right)}$$

Multiply 0.004 times -112.

$$x=\frac{-100.02±\sqrt{10004.0004-0.448}}{2\left(-0.001\right)}$$

Add 10004.0004 to -0.448 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-100.02±\sqrt{10003.5524}}{2\left(-0.001\right)}$$

Take the square root of 10003.5524.

$$x=\frac{-100.02±\frac{\sqrt{25008881}}{50}}{2\left(-0.001\right)}$$

Multiply 2 times -0.001.

$$x=\frac{-100.02±\frac{\sqrt{25008881}}{50}}{-0.002}$$

Now solve the equation x=\frac{-100.02±\frac{\sqrt{25008881}}{50}}{-0.002} when ± is plus. Add -100.02 to \frac{\sqrt{25008881}}{50}.

$$x=\frac{\sqrt{25008881}-5001}{-0.002\times 50}$$

Divide \frac{-5001+\sqrt{25008881}}{50} by -0.002 by multiplying \frac{-5001+\sqrt{25008881}}{50} by the reciprocal of -0.002.

$$x=50010-10\sqrt{25008881}$$

Now solve the equation x=\frac{-100.02±\frac{\sqrt{25008881}}{50}}{-0.002} when ± is minus. Subtract \frac{\sqrt{25008881}}{50} from -100.02.

$$x=\frac{-\sqrt{25008881}-5001}{-0.002\times 50}$$

Divide \frac{-5001-\sqrt{25008881}}{50} by -0.002 by multiplying \frac{-5001-\sqrt{25008881}}{50} by the reciprocal of -0.002.

$$x=10\sqrt{25008881}+50010$$

The equation is now solved.

$$x=50010-10\sqrt{25008881} x=10\sqrt{25008881}+50010$$

Steps for Completing the Square

Use the distributive property to multiply 0.00001 by 20-x.

$$112=100x\left(1+0.0002-0.00001x\right)$$

Add 1 and 0.0002 to get 1.0002.

$$112=100x\left(1.0002-0.00001x\right)$$

Use the distributive property to multiply 100x by 1.0002-0.00001x.

$$112=100.02x-0.001x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$100.02x-0.001x^{2}=112$$

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.

$$-0.001x^{2}+100.02x=112$$

Multiply both sides by -1000.

$$\frac{-0.001x^{2}+100.02x}{-0.001}=\frac{112}{-0.001}$$

Dividing by -0.001 undoes the multiplication by -0.001.

$$x^{2}+\frac{100.02}{-0.001}x=\frac{112}{-0.001}$$

Divide 100.02 by -0.001 by multiplying 100.02 by the reciprocal of -0.001.

$$x^{2}-100020x=\frac{112}{-0.001}$$

Divide 112 by -0.001 by multiplying 112 by the reciprocal of -0.001.

$$x^{2}-100020x=-112000$$

Divide -100020, the coefficient of the x term, by 2 to get -50010. Then add the square of -50010 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100020x+\left(-50010\right)^{2}=-112000+\left(-50010\right)^{2}$$

Square -50010.

$$x^{2}-100020x+2501000100=-112000+2501000100$$

Add -112000 to 2501000100.

$$x^{2}-100020x+2501000100=2500888100$$

Factor x^{2}-100020x+2501000100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50010\right)^{2}=2500888100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50010\right)^{2}}=\sqrt{2500888100}$$

Simplify.

$$x-50010=10\sqrt{25008881} x-50010=-10\sqrt{25008881}$$

Add 50010 to both sides of the equation.

$$x=10\sqrt{25008881}+50010 x=50010-10\sqrt{25008881}$$