Evaluate
7n^{2}-8n+9
Differentiate w.r.t. n
14n-8
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x ^ 2 -\frac{8}{7}x +\frac{9}{7} = 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 7
r + s = \frac{8}{7} rs = \frac{9}{7}
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{4}{7} - u s = \frac{4}{7} + u
Two numbers r and s sum up to \frac{8}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{8}{7} = \frac{4}{7}. 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{4}{7} - u) (\frac{4}{7} + u) = \frac{9}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{7}
\frac{16}{49} - u^2 = \frac{9}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{9}{7}-\frac{16}{49} = \frac{47}{49}
Simplify the expression by subtracting \frac{16}{49} on both sides
u^2 = -\frac{47}{49} u = \pm\sqrt{-\frac{47}{49}} = \pm \frac{\sqrt{47}}{7}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{4}{7} - \frac{\sqrt{47}}{7}i = 0.571 - 0.979i s = \frac{4}{7} + \frac{\sqrt{47}}{7}i = 0.571 + 0.979i
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 7n^{2-1}-8n^{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}.
14n^{2-1}-8n^{1-1}
Multiply 2 times 7.
14n^{1}-8n^{1-1}
Subtract 1 from 2.
14n^{1}-8n^{0}
Subtract 1 from 1.
14n-8n^{0}
For any term t, t^{1}=t.
14n-8
For any term t except 0, t^{0}=1.
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}