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81x^{2}-81x=88
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.
81x^{2}-81x-88=88-88
Subtract 88 from both sides of the equation.
81x^{2}-81x-88=0
Subtracting 88 from itself leaves 0.
x=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 81\left(-88\right)}}{2\times 81}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, -81 for b, and -88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-81\right)±\sqrt{6561-4\times 81\left(-88\right)}}{2\times 81}
Square -81.
x=\frac{-\left(-81\right)±\sqrt{6561-324\left(-88\right)}}{2\times 81}
Multiply -4 times 81.
x=\frac{-\left(-81\right)±\sqrt{6561+28512}}{2\times 81}
Multiply -324 times -88.
x=\frac{-\left(-81\right)±\sqrt{35073}}{2\times 81}
Add 6561 to 28512.
x=\frac{-\left(-81\right)±9\sqrt{433}}{2\times 81}
Take the square root of 35073.
x=\frac{81±9\sqrt{433}}{2\times 81}
The opposite of -81 is 81.
x=\frac{81±9\sqrt{433}}{162}
Multiply 2 times 81.
x=\frac{9\sqrt{433}+81}{162}
Now solve the equation x=\frac{81±9\sqrt{433}}{162} when ± is plus. Add 81 to 9\sqrt{433}.
x=\frac{\sqrt{433}}{18}+\frac{1}{2}
Divide 81+9\sqrt{433} by 162.
x=\frac{81-9\sqrt{433}}{162}
Now solve the equation x=\frac{81±9\sqrt{433}}{162} when ± is minus. Subtract 9\sqrt{433} from 81.
x=-\frac{\sqrt{433}}{18}+\frac{1}{2}
Divide 81-9\sqrt{433} by 162.
x=\frac{\sqrt{433}}{18}+\frac{1}{2} x=-\frac{\sqrt{433}}{18}+\frac{1}{2}
The equation is now solved.
81x^{2}-81x=88
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{81x^{2}-81x}{81}=\frac{88}{81}
Divide both sides by 81.
x^{2}+\left(-\frac{81}{81}\right)x=\frac{88}{81}
Dividing by 81 undoes the multiplication by 81.
x^{2}-x=\frac{88}{81}
Divide -81 by 81.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{88}{81}+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=\frac{88}{81}+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{433}{324}
Add \frac{88}{81} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{2}\right)^{2}=\frac{433}{324}
Factor x^{2}-x+\frac{1}{4}. 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{1}{2}\right)^{2}}=\sqrt{\frac{433}{324}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{433}}{18} x-\frac{1}{2}=-\frac{\sqrt{433}}{18}
Simplify.
x=\frac{\sqrt{433}}{18}+\frac{1}{2} x=-\frac{\sqrt{433}}{18}+\frac{1}{2}
Add \frac{1}{2} to both sides of the equation.