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\left(1000+1000x\right)\left(0.98+x\right)=1000+108
Use the distributive property to multiply 1000 by 1+x.
980+1980x+1000x^{2}=1000+108
Use the distributive property to multiply 1000+1000x by 0.98+x and combine like terms.
980+1980x+1000x^{2}=1108
Add 1000 and 108 to get 1108.
980+1980x+1000x^{2}-1108=0
Subtract 1108 from both sides.
-128+1980x+1000x^{2}=0
Subtract 1108 from 980 to get -128.
1000x^{2}+1980x-128=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.
x=\frac{-1980±\sqrt{1980^{2}-4\times 1000\left(-128\right)}}{2\times 1000}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, 1980 for b, and -128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1980±\sqrt{3920400-4\times 1000\left(-128\right)}}{2\times 1000}
Square 1980.
x=\frac{-1980±\sqrt{3920400-4000\left(-128\right)}}{2\times 1000}
Multiply -4 times 1000.
x=\frac{-1980±\sqrt{3920400+512000}}{2\times 1000}
Multiply -4000 times -128.
x=\frac{-1980±\sqrt{4432400}}{2\times 1000}
Add 3920400 to 512000.
x=\frac{-1980±20\sqrt{11081}}{2\times 1000}
Take the square root of 4432400.
x=\frac{-1980±20\sqrt{11081}}{2000}
Multiply 2 times 1000.
x=\frac{20\sqrt{11081}-1980}{2000}
Now solve the equation x=\frac{-1980±20\sqrt{11081}}{2000} when ± is plus. Add -1980 to 20\sqrt{11081}.
x=\frac{\sqrt{11081}-99}{100}
Divide -1980+20\sqrt{11081} by 2000.
x=\frac{-20\sqrt{11081}-1980}{2000}
Now solve the equation x=\frac{-1980±20\sqrt{11081}}{2000} when ± is minus. Subtract 20\sqrt{11081} from -1980.
x=\frac{-\sqrt{11081}-99}{100}
Divide -1980-20\sqrt{11081} by 2000.
x=\frac{\sqrt{11081}-99}{100} x=\frac{-\sqrt{11081}-99}{100}
The equation is now solved.
\left(1000+1000x\right)\left(0.98+x\right)=1000+108
Use the distributive property to multiply 1000 by 1+x.
980+1980x+1000x^{2}=1000+108
Use the distributive property to multiply 1000+1000x by 0.98+x and combine like terms.
980+1980x+1000x^{2}=1108
Add 1000 and 108 to get 1108.
1980x+1000x^{2}=1108-980
Subtract 980 from both sides.
1980x+1000x^{2}=128
Subtract 980 from 1108 to get 128.
1000x^{2}+1980x=128
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.
\frac{1000x^{2}+1980x}{1000}=\frac{128}{1000}
Divide both sides by 1000.
x^{2}+\frac{1980}{1000}x=\frac{128}{1000}
Dividing by 1000 undoes the multiplication by 1000.
x^{2}+\frac{99}{50}x=\frac{128}{1000}
Reduce the fraction \frac{1980}{1000} to lowest terms by extracting and canceling out 20.
x^{2}+\frac{99}{50}x=\frac{16}{125}
Reduce the fraction \frac{128}{1000} to lowest terms by extracting and canceling out 8.
x^{2}+\frac{99}{50}x+\left(\frac{99}{100}\right)^{2}=\frac{16}{125}+\left(\frac{99}{100}\right)^{2}
Divide \frac{99}{50}, the coefficient of the x term, by 2 to get \frac{99}{100}. Then add the square of \frac{99}{100} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{99}{50}x+\frac{9801}{10000}=\frac{16}{125}+\frac{9801}{10000}
Square \frac{99}{100} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{99}{50}x+\frac{9801}{10000}=\frac{11081}{10000}
Add \frac{16}{125} to \frac{9801}{10000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{99}{100}\right)^{2}=\frac{11081}{10000}
Factor x^{2}+\frac{99}{50}x+\frac{9801}{10000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{99}{100}\right)^{2}}=\sqrt{\frac{11081}{10000}}
Take the square root of both sides of the equation.
x+\frac{99}{100}=\frac{\sqrt{11081}}{100} x+\frac{99}{100}=-\frac{\sqrt{11081}}{100}
Simplify.
x=\frac{\sqrt{11081}-99}{100} x=\frac{-\sqrt{11081}-99}{100}
Subtract \frac{99}{100} from both sides of the equation.