Evaluate
\frac{19x^{4}}{48}-4x^{3}+\frac{9x^{2}}{2}+206
Expand
\frac{19x^{4}}{48}-4x^{3}+\frac{9x^{2}}{2}+206
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\left(3x-\left(\frac{1}{4}x^{2}+3x+9\right)\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{2}x+3\right)^{2}.
\left(3x-\frac{1}{4}x^{2}-3x-9\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
To find the opposite of \frac{1}{4}x^{2}+3x+9, find the opposite of each term.
\left(-\frac{1}{4}x^{2}-9\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Combine 3x and -3x to get 0.
\frac{1}{16}\left(x^{2}\right)^{2}+\frac{9}{2}x^{2}+81+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-\frac{1}{4}x^{2}-9\right)^{2}.
\frac{1}{16}x^{4}+\frac{9}{2}x^{2}+81+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
\frac{1}{16}x^{4}+\frac{9}{2}x^{2}+81+\frac{1}{3}x^{4}-4x^{3}-\left(-5\right)^{3}
Use the distributive property to multiply \frac{2}{3}x^{3} by \frac{1}{2}x-6.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}-\left(-5\right)^{3}
Combine \frac{1}{16}x^{4} and \frac{1}{3}x^{4} to get \frac{19}{48}x^{4}.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}-\left(-125\right)
Calculate -5 to the power of 3 and get -125.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}+125
The opposite of -125 is 125.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+206-4x^{3}
Add 81 and 125 to get 206.
\left(3x-\left(\frac{1}{4}x^{2}+3x+9\right)\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{2}x+3\right)^{2}.
\left(3x-\frac{1}{4}x^{2}-3x-9\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
To find the opposite of \frac{1}{4}x^{2}+3x+9, find the opposite of each term.
\left(-\frac{1}{4}x^{2}-9\right)^{2}+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Combine 3x and -3x to get 0.
\frac{1}{16}\left(x^{2}\right)^{2}+\frac{9}{2}x^{2}+81+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-\frac{1}{4}x^{2}-9\right)^{2}.
\frac{1}{16}x^{4}+\frac{9}{2}x^{2}+81+\frac{2}{3}x^{3}\left(\frac{1}{2}x-6\right)-\left(-5\right)^{3}
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
\frac{1}{16}x^{4}+\frac{9}{2}x^{2}+81+\frac{1}{3}x^{4}-4x^{3}-\left(-5\right)^{3}
Use the distributive property to multiply \frac{2}{3}x^{3} by \frac{1}{2}x-6.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}-\left(-5\right)^{3}
Combine \frac{1}{16}x^{4} and \frac{1}{3}x^{4} to get \frac{19}{48}x^{4}.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}-\left(-125\right)
Calculate -5 to the power of 3 and get -125.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+81-4x^{3}+125
The opposite of -125 is 125.
\frac{19}{48}x^{4}+\frac{9}{2}x^{2}+206-4x^{3}
Add 81 and 125 to get 206.
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}