Evaluate
4
Factor
2^{2}
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2+\sqrt{5}-\frac{2-\sqrt{5}}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}
Rationalize the denominator of \frac{1}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.
2+\sqrt{5}-\frac{2-\sqrt{5}}{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}.
2+\sqrt{5}-\frac{2-\sqrt{5}}{4-5}
Square 2. Square \sqrt{5}.
2+\sqrt{5}-\frac{2-\sqrt{5}}{-1}
Subtract 5 from 4 to get -1.
2+\sqrt{5}-\left(-2-\left(-\sqrt{5}\right)\right)
Anything divided by -1 gives its opposite. To find the opposite of 2-\sqrt{5}, find the opposite of each term.
2+\sqrt{5}-\left(-2\right)-\left(-\left(-\sqrt{5}\right)\right)
To find the opposite of -2-\left(-\sqrt{5}\right), find the opposite of each term.
2+\sqrt{5}+2-\left(-\left(-\sqrt{5}\right)\right)
The opposite of -2 is 2.
2+\sqrt{5}+2-\sqrt{5}
The opposite of -\sqrt{5} is \sqrt{5}.
4+\sqrt{5}-\sqrt{5}
Add 2 and 2 to get 4.
4
Subtract \sqrt{5} from \sqrt{5} to get 0.
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}