Solve 22*10^5=10^5+1000/20*10^-4(22-x)x

$$22 \times 10 ^ { 5 } = 10 ^ { 5 } + \frac { 1000 } { 20 \times 10 ^ { - 4 } } ( 22 - x ) x$$

Solve for x

$$x=\frac{2\sqrt{730}}{5}+11\approx 21.807404869$$

$$x=-\frac{2\sqrt{730}}{5}+11\approx 0.192595131$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 5 and get 100000.

$$22\times 100000=10^{5}+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Multiply 22 and 100000 to get 2200000.

$$2200000=10^{5}+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Calculate 10 to the power of 5 and get 100000.

$$2200000=100000+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$2200000=100000+\frac{1000}{20\times \frac{1}{10000}}\left(22-x\right)x$$

Multiply 20 and \frac{1}{10000} to get \frac{1}{500}.

$$2200000=100000+\frac{1000}{\frac{1}{500}}\left(22-x\right)x$$

Divide 1000 by \frac{1}{500} by multiplying 1000 by the reciprocal of \frac{1}{500}.

$$2200000=100000+1000\times 500\left(22-x\right)x$$

Multiply 1000 and 500 to get 500000.

$$2200000=100000+500000\left(22-x\right)x$$

Use the distributive property to multiply 500000 by 22-x.

$$2200000=100000+\left(11000000-500000x\right)x$$

Use the distributive property to multiply 11000000-500000x by x.

$$2200000=100000+11000000x-500000x^{2}$$

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

$$100000+11000000x-500000x^{2}=2200000$$

Subtract 2200000 from both sides.

$$100000+11000000x-500000x^{2}-2200000=0$$

Subtract 2200000 from 100000 to get -2100000.

$$-2100000+11000000x-500000x^{2}=0$$

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.

$$-500000x^{2}[phone]x-2100000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -500000 for a, 11000000 for b, and -2100000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11000000±\sqrt{11000000^{2}-4\left(-500000\right)\left(-2100000\right)}}{2\left(-500000\right)}$$

Square 11000000.

$$x=\frac{-11000000±\sqrt{121000000000000-4\left(-500000\right)\left(-2100000\right)}}{2\left(-500000\right)}$$

Multiply -4 times -500000.

$$x=\frac{-11000000±\sqrt{121000000000000+2000000\left(-2100000\right)}}{2\left(-500000\right)}$$

Multiply 2000000 times -2100000.

$$x=\frac{-11000000±\sqrt{121000000000000-4200000000000}}{2\left(-500000\right)}$$

Add 121000000000000 to -4200000000000.

$$x=\frac{-11000000±\sqrt{116800000000000}}{2\left(-500000\right)}$$

Take the square root of 116800000000000.

$$x=\frac{-11000000±400000\sqrt{730}}{2\left(-500000\right)}$$

Multiply 2 times -500000.

$$x=\frac{-11000000±400000\sqrt{730}}{-1000000}$$

Now solve the equation x=\frac{-11000000±400000\sqrt{730}}{-1000000} when ± is plus. Add -11000000 to 400000\sqrt{730}.

$$x=\frac{400000\sqrt{730}-11000000}{-1000000}$$

Divide -11000000+400000\sqrt{730} by -1000000.

$$x=-\frac{2\sqrt{730}}{5}+11$$

Now solve the equation x=\frac{-11000000±400000\sqrt{730}}{-1000000} when ± is minus. Subtract 400000\sqrt{730} from -11000000.

$$x=\frac{-400000\sqrt{730}-11000000}{-1000000}$$

Divide -11000000-400000\sqrt{730} by -1000000.

$$x=\frac{2\sqrt{730}}{5}+11$$

The equation is now solved.

$$x=-\frac{2\sqrt{730}}{5}+11 x=\frac{2\sqrt{730}}{5}+11$$

Steps for Completing the Square

Calculate 10 to the power of 5 and get 100000.

$$22\times 100000=10^{5}+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Multiply 22 and 100000 to get 2200000.

$$2200000=10^{5}+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Calculate 10 to the power of 5 and get 100000.

$$2200000=100000+\frac{1000}{20\times 10^{-4}}\left(22-x\right)x$$

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$2200000=100000+\frac{1000}{20\times \frac{1}{10000}}\left(22-x\right)x$$

Multiply 20 and \frac{1}{10000} to get \frac{1}{500}.

$$2200000=100000+\frac{1000}{\frac{1}{500}}\left(22-x\right)x$$

Divide 1000 by \frac{1}{500} by multiplying 1000 by the reciprocal of \frac{1}{500}.

$$2200000=100000+1000\times 500\left(22-x\right)x$$

Multiply 1000 and 500 to get 500000.

$$2200000=100000+500000\left(22-x\right)x$$

Use the distributive property to multiply 500000 by 22-x.

$$2200000=100000+\left(11000000-500000x\right)x$$

Use the distributive property to multiply 11000000-500000x by x.

$$2200000=100000+11000000x-500000x^{2}$$

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

$$100000+11000000x-500000x^{2}=2200000$$

Subtract 100000 from both sides.

$$11000000x-500000x^{2}=2200000-100000$$

Subtract 100000 from 2200000 to get 2100000.

$$11000000x-500000x^{2}=2100000$$

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.

$$-500000x^{2}[phone]x=2100000$$

Divide both sides by -500000.

$$\frac{-500000x^{2}[phone]x}{-500000}=\frac{2100000}{-500000}$$

Dividing by -500000 undoes the multiplication by -500000.

$$x^{2}+\frac{11000000}{-500000}x=\frac{2100000}{-500000}$$

Divide 11000000 by -500000.

$$x^{2}-22x=\frac{2100000}{-500000}$$

Reduce the fraction \frac{2100000}{-500000} to lowest terms by extracting and canceling out 100000.

$$x^{2}-22x=-\frac{21}{5}$$

Divide -22, the coefficient of the x term, by 2 to get -11. Then add the square of -11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-22x+\left(-11\right)^{2}=-\frac{21}{5}+\left(-11\right)^{2}$$

Square -11.

$$x^{2}-22x+121=-\frac{21}{5}+121$$

Add -\frac{21}{5} to 121.

$$x^{2}-22x+121=\frac{584}{5}$$

Factor x^{2}-22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-11\right)^{2}=\frac{584}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{\frac{584}{5}}$$

Simplify.

$$x-11=\frac{2\sqrt{730}}{5} x-11=-\frac{2\sqrt{730}}{5}$$

Add 11 to both sides of the equation.

$$x=\frac{2\sqrt{730}}{5}+11 x=-\frac{2\sqrt{730}}{5}+11$$