Evaluate
1+4x-2x^{2}
Factor
-2\left(x-\left(-\frac{\sqrt{6}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{6}}{2}+1\right)\right)
Graph
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-2x^{2}-4x\left(-1\right)+1
Multiply 2 and -1 to get -2.
-2x^{2}-\left(-4x\right)+1
Multiply 4 and -1 to get -4.
-2x^{2}+4x+1
The opposite of -4x is 4x.
factor(-2x^{2}-4x\left(-1\right)+1)
Multiply 2 and -1 to get -2.
factor(-2x^{2}-\left(-4x\right)+1)
Multiply 4 and -1 to get -4.
factor(-2x^{2}+4x+1)
The opposite of -4x is 4x.
-2x^{2}+4x+1=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)}}{2\left(-2\right)}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-4±\sqrt{16-4\left(-2\right)}}{2\left(-2\right)}
Square 4.
x=\frac{-4±\sqrt{16+8}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-4±\sqrt{24}}{2\left(-2\right)}
Add 16 to 8.
x=\frac{-4±2\sqrt{6}}{2\left(-2\right)}
Take the square root of 24.
x=\frac{-4±2\sqrt{6}}{-4}
Multiply 2 times -2.
x=\frac{2\sqrt{6}-4}{-4}
Now solve the equation x=\frac{-4±2\sqrt{6}}{-4} when ± is plus. Add -4 to 2\sqrt{6}.
x=-\frac{\sqrt{6}}{2}+1
Divide -4+2\sqrt{6} by -4.
x=\frac{-2\sqrt{6}-4}{-4}
Now solve the equation x=\frac{-4±2\sqrt{6}}{-4} when ± is minus. Subtract 2\sqrt{6} from -4.
x=\frac{\sqrt{6}}{2}+1
Divide -4-2\sqrt{6} by -4.
-2x^{2}+4x+1=-2\left(x-\left(-\frac{\sqrt{6}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{6}}{2}+1\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{\sqrt{6}}{2} for x_{1} and 1+\frac{\sqrt{6}}{2} for x_{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}