Evaluate
6x^{2}+2x+1
Differentiate w.r.t. x
12x+2
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x ^ 2 +\frac{1}{3}x +\frac{1}{6} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 6
r + s = -\frac{1}{3} rs = \frac{1}{6}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{1}{6} - u s = -\frac{1}{6} + u
Two numbers r and s sum up to -\frac{1}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{3} = -\frac{1}{6}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{1}{6} - u) (-\frac{1}{6} + u) = \frac{1}{6}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{6}
\frac{1}{36} - u^2 = \frac{1}{6}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{6}-\frac{1}{36} = \frac{5}{36}
Simplify the expression by subtracting \frac{1}{36} on both sides
u^2 = -\frac{5}{36} u = \pm\sqrt{-\frac{5}{36}} = \pm \frac{\sqrt{5}}{6}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{1}{6} - \frac{\sqrt{5}}{6}i = -0.167 - 0.373i s = -\frac{1}{6} + \frac{\sqrt{5}}{6}i = -0.167 + 0.373i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
2\times 6x^{2-1}+2x^{1-1}
The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.
12x^{2-1}+2x^{1-1}
Multiply 2 times 6.
12x^{1}+2x^{1-1}
Subtract 1 from 2.
12x^{1}+2x^{0}
Subtract 1 from 1.
12x+2x^{0}
For any term t, t^{1}=t.
12x+2\times 1
For any term t except 0, t^{0}=1.
12x+2
For any term t, t\times 1=t and 1t=t.
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}