Evaluate
\frac{10-15\sqrt{5}}{41}\approx -0.574171211
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\frac{-5\left(3\sqrt{5}-2\right)}{\left(3\sqrt{5}+2\right)\left(3\sqrt{5}-2\right)}
Rationalize the denominator of \frac{-5}{3\sqrt{5}+2} by multiplying numerator and denominator by 3\sqrt{5}-2.
\frac{-5\left(3\sqrt{5}-2\right)}{\left(3\sqrt{5}\right)^{2}-2^{2}}
Consider \left(3\sqrt{5}+2\right)\left(3\sqrt{5}-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{-5\left(3\sqrt{5}-2\right)}{3^{2}\left(\sqrt{5}\right)^{2}-2^{2}}
Expand \left(3\sqrt{5}\right)^{2}.
\frac{-5\left(3\sqrt{5}-2\right)}{9\left(\sqrt{5}\right)^{2}-2^{2}}
Calculate 3 to the power of 2 and get 9.
\frac{-5\left(3\sqrt{5}-2\right)}{9\times 5-2^{2}}
The square of \sqrt{5} is 5.
\frac{-5\left(3\sqrt{5}-2\right)}{45-2^{2}}
Multiply 9 and 5 to get 45.
\frac{-5\left(3\sqrt{5}-2\right)}{45-4}
Calculate 2 to the power of 2 and get 4.
\frac{-5\left(3\sqrt{5}-2\right)}{41}
Subtract 4 from 45 to get 41.
\frac{-15\sqrt{5}+10}{41}
Use the distributive property to multiply -5 by 3\sqrt{5}-2.
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}