Solve 1*1.01(42-x)*0.99x=0.8

$$1 \cdot 1.01 ( 42 - x ) \cdot 0.99 x = 0.8$$

Solve for x

$$x=\frac{\sqrt{4890132049}}{3333}+21\approx 41.980941828$$

$$x=-\frac{\sqrt{4890132049}}{3333}+21\approx 0.019058172$$

Steps Using the Quadratic Formula

Multiply 1 and 1.01 to get 1.01.

$$1.01\left(42-x\right)\times 0.99x=0.8$$

Multiply 1.01 and 0.99 to get 0.9999.

$$0.9999\left(42-x\right)x=0.8$$

Use the distributive property to multiply 0.9999 by 42-x.

$$\left(41.9958-0.9999x\right)x=0.8$$

Use the distributive property to multiply 41.9958-0.9999x by x.

$$41.9958x-0.9999x^{2}=0.8$$

Subtract 0.8 from both sides.

$$41.9958x-0.9999x^{2}-0.8=0$$

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.

$$-0.9999x^{2}+41.9958x-0.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.9999 for a, 41.9958 for b, and -0.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-41.9958±\sqrt{41.9958^{2}-4\left(-0.9999\right)\left(-0.8\right)}}{2\left(-0.9999\right)}$$

Square 41.9958 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-41.9958±\sqrt{1763.64721764-4\left(-0.9999\right)\left(-0.8\right)}}{2\left(-0.9999\right)}$$

Multiply -4 times -0.9999.

$$x=\frac{-41.9958±\sqrt{1763.64721764+3.9996\left(-0.8\right)}}{2\left(-0.9999\right)}$$

Multiply 3.9996 times -0.8 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-41.9958±\sqrt{1763.64721764-3.19968}}{2\left(-0.9999\right)}$$

Add 1763.64721764 to -3.19968 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-41.9958±\sqrt{1760.44753764}}{2\left(-0.9999\right)}$$

Take the square root of 1760.44753764.

$$x=\frac{-41.9958±\frac{3\sqrt{4890132049}}{5000}}{2\left(-0.9999\right)}$$

Multiply 2 times -0.9999.

$$x=\frac{-41.9958±\frac{3\sqrt{4890132049}}{5000}}{-1.9998}$$

Now solve the equation x=\frac{-41.9958±\frac{3\sqrt{4890132049}}{5000}}{-1.9998} when ± is plus. Add -41.9958 to \frac{3\sqrt{4890132049}}{5000}.

$$x=\frac{3\sqrt{4890132049}-209979}{-1.9998\times 5000}$$

Divide \frac{-209979+3\sqrt{4890132049}}{5000} by -1.9998 by multiplying \frac{-209979+3\sqrt{4890132049}}{5000} by the reciprocal of -1.9998.

$$x=-\frac{\sqrt{4890132049}}{3333}+21$$

Now solve the equation x=\frac{-41.9958±\frac{3\sqrt{4890132049}}{5000}}{-1.9998} when ± is minus. Subtract \frac{3\sqrt{4890132049}}{5000} from -41.9958.

$$x=\frac{-3\sqrt{4890132049}-209979}{-1.9998\times 5000}$$

Divide \frac{-209979-3\sqrt{4890132049}}{5000} by -1.9998 by multiplying \frac{-209979-3\sqrt{4890132049}}{5000} by the reciprocal of -1.9998.

$$x=\frac{\sqrt{4890132049}}{3333}+21$$

The equation is now solved.

$$x=-\frac{\sqrt{4890132049}}{3333}+21 x=\frac{\sqrt{4890132049}}{3333}+21$$

Steps for Completing the Square

Multiply 1 and 1.01 to get 1.01.

$$1.01\left(42-x\right)\times 0.99x=0.8$$

Multiply 1.01 and 0.99 to get 0.9999.

$$0.9999\left(42-x\right)x=0.8$$

Use the distributive property to multiply 0.9999 by 42-x.

$$\left(41.9958-0.9999x\right)x=0.8$$

Use the distributive property to multiply 41.9958-0.9999x by x.

$$41.9958x-0.9999x^{2}=0.8$$

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.

$$-0.9999x^{2}+41.9958x=0.8$$

Divide both sides of the equation by -0.9999, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.9999x^{2}+41.9958x}{-0.9999}=\frac{0.8}{-0.9999}$$

Dividing by -0.9999 undoes the multiplication by -0.9999.

$$x^{2}+\frac{41.9958}{-0.9999}x=\frac{0.8}{-0.9999}$$

Divide 41.9958 by -0.9999 by multiplying 41.9958 by the reciprocal of -0.9999.

$$x^{2}-42x=\frac{0.8}{-0.9999}$$

Divide 0.8 by -0.9999 by multiplying 0.8 by the reciprocal of -0.9999.

$$x^{2}-42x=-\frac{8000}{9999}$$

Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-42x+\left(-21\right)^{2}=-\frac{8000}{9999}+\left(-21\right)^{2}$$

Square -21.

$$x^{2}-42x+441=-\frac{8000}{9999}+441$$

Add -\frac{8000}{9999} to 441.

$$x^{2}-42x+441=\frac{4401559}{9999}$$

Factor x^{2}-42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-21\right)^{2}=\frac{4401559}{9999}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-21\right)^{2}}=\sqrt{\frac{4401559}{9999}}$$

Simplify.

$$x-21=\frac{\sqrt{4890132049}}{3333} x-21=-\frac{\sqrt{4890132049}}{3333}$$

Add 21 to both sides of the equation.

$$x=\frac{\sqrt{4890132049}}{3333}+21 x=-\frac{\sqrt{4890132049}}{3333}+21$$