Solve for x
x=9\sqrt{3}+18\approx 33.588457268
x=18-9\sqrt{3}\approx 2.411542732
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-x^{2}+36x-81=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{-36±\sqrt{36^{2}-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 36 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-36±\sqrt{1296-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}
Square 36.
x=\frac{-36±\sqrt{1296+4\left(-81\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-36±\sqrt{1296-324}}{2\left(-1\right)}
Multiply 4 times -81.
x=\frac{-36±\sqrt{972}}{2\left(-1\right)}
Add 1296 to -324.
x=\frac{-36±18\sqrt{3}}{2\left(-1\right)}
Take the square root of 972.
x=\frac{-36±18\sqrt{3}}{-2}
Multiply 2 times -1.
x=\frac{18\sqrt{3}-36}{-2}
Now solve the equation x=\frac{-36±18\sqrt{3}}{-2} when ± is plus. Add -36 to 18\sqrt{3}.
x=18-9\sqrt{3}
Divide -36+18\sqrt{3} by -2.
x=\frac{-18\sqrt{3}-36}{-2}
Now solve the equation x=\frac{-36±18\sqrt{3}}{-2} when ± is minus. Subtract 18\sqrt{3} from -36.
x=9\sqrt{3}+18
Divide -36-18\sqrt{3} by -2.
x=18-9\sqrt{3} x=9\sqrt{3}+18
The equation is now solved.
-x^{2}+36x-81=0
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.
-x^{2}+36x-81-\left(-81\right)=-\left(-81\right)
Add 81 to both sides of the equation.
-x^{2}+36x=-\left(-81\right)
Subtracting -81 from itself leaves 0.
-x^{2}+36x=81
Subtract -81 from 0.
\frac{-x^{2}+36x}{-1}=\frac{81}{-1}
Divide both sides by -1.
x^{2}+\frac{36}{-1}x=\frac{81}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-36x=\frac{81}{-1}
Divide 36 by -1.
x^{2}-36x=-81
Divide 81 by -1.
x^{2}-36x+\left(-18\right)^{2}=-81+\left(-18\right)^{2}
Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-36x+324=-81+324
Square -18.
x^{2}-36x+324=243
Add -81 to 324.
\left(x-18\right)^{2}=243
Factor x^{2}-36x+324. 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-18\right)^{2}}=\sqrt{243}
Take the square root of both sides of the equation.
x-18=9\sqrt{3} x-18=-9\sqrt{3}
Simplify.
x=9\sqrt{3}+18 x=18-9\sqrt{3}
Add 18 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}