Evaluate
3\left(\sqrt{5}-2\right)\approx 0.708203932
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\frac{3\left(2-\sqrt{5}\right)}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}
Rationalize the denominator of \frac{3}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.
\frac{3\left(2-\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}
Consider \left(2+\sqrt{5}\right)\left(2-\sqrt{5}\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{3\left(2-\sqrt{5}\right)}{4-5}
Square 2. Square \sqrt{5}.
\frac{3\left(2-\sqrt{5}\right)}{-1}
Subtract 5 from 4 to get -1.
-3\left(2-\sqrt{5}\right)
Anything divided by -1 gives its opposite.
-6+3\sqrt{5}
Use the distributive property to multiply -3 by 2-\sqrt{5}.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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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}