Solve for x
x=\sqrt{5}+3\approx 5.236067977
x=3-\sqrt{5}\approx 0.763932023
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72=3x\left(-6x+36\right)
Multiply both sides of the equation by 2.
72=-18x^{2}+108x
Use the distributive property to multiply 3x by -6x+36.
-18x^{2}+108x=72
Swap sides so that all variable terms are on the left hand side.
-18x^{2}+108x-72=0
Subtract 72 from both sides.
x=\frac{-108±\sqrt{108^{2}-4\left(-18\right)\left(-72\right)}}{2\left(-18\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 108 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-108±\sqrt{11664-4\left(-18\right)\left(-72\right)}}{2\left(-18\right)}
Square 108.
x=\frac{-108±\sqrt{11664+72\left(-72\right)}}{2\left(-18\right)}
Multiply -4 times -18.
x=\frac{-108±\sqrt{11664-5184}}{2\left(-18\right)}
Multiply 72 times -72.
x=\frac{-108±\sqrt{6480}}{2\left(-18\right)}
Add 11664 to -5184.
x=\frac{-108±36\sqrt{5}}{2\left(-18\right)}
Take the square root of 6480.
x=\frac{-108±36\sqrt{5}}{-36}
Multiply 2 times -18.
x=\frac{36\sqrt{5}-108}{-36}
Now solve the equation x=\frac{-108±36\sqrt{5}}{-36} when ± is plus. Add -108 to 36\sqrt{5}.
x=3-\sqrt{5}
Divide -108+36\sqrt{5} by -36.
x=\frac{-36\sqrt{5}-108}{-36}
Now solve the equation x=\frac{-108±36\sqrt{5}}{-36} when ± is minus. Subtract 36\sqrt{5} from -108.
x=\sqrt{5}+3
Divide -108-36\sqrt{5} by -36.
x=3-\sqrt{5} x=\sqrt{5}+3
The equation is now solved.
72=3x\left(-6x+36\right)
Multiply both sides of the equation by 2.
72=-18x^{2}+108x
Use the distributive property to multiply 3x by -6x+36.
-18x^{2}+108x=72
Swap sides so that all variable terms are on the left hand side.
\frac{-18x^{2}+108x}{-18}=\frac{72}{-18}
Divide both sides by -18.
x^{2}+\frac{108}{-18}x=\frac{72}{-18}
Dividing by -18 undoes the multiplication by -18.
x^{2}-6x=\frac{72}{-18}
Divide 108 by -18.
x^{2}-6x=-4
Divide 72 by -18.
x^{2}-6x+\left(-3\right)^{2}=-4+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=-4+9
Square -3.
x^{2}-6x+9=5
Add -4 to 9.
\left(x-3\right)^{2}=5
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{5}
Take the square root of both sides of the equation.
x-3=\sqrt{5} x-3=-\sqrt{5}
Simplify.
x=\sqrt{5}+3 x=3-\sqrt{5}
Add 3 to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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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}