Evaluate

3x^{2}+2x+9

$3x_{2}+2x+9$

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x ^ 2 +\frac{2}{3}x +3 = 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 3

r + s = -\frac{2}{3} rs = 3

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}{3} - u s = -\frac{1}{3} + u

Two numbers r and s sum up to -\frac{2}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{3} = -\frac{1}{3}. 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}{3} - u) (-\frac{1}{3} + u) = 3

To solve for unknown quantity u, substitute these in the product equation rs = 3

\frac{1}{9} - u^2 = 3

Simplify by expanding (a -b) (a + b) = a^2 – b^2

-u^2 = 3-\frac{1}{9} = \frac{26}{9}

Simplify the expression by subtracting \frac{1}{9} on both sides

u^2 = -\frac{26}{9} u = \pm\sqrt{-\frac{26}{9}} = \pm \frac{\sqrt{26}}{3}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}{3} - \frac{\sqrt{26}}{3}i = -0.333 - 1.700i s = -\frac{1}{3} + \frac{\sqrt{26}}{3}i = -0.333 + 1.700i

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

Examples

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$x_{2}−4x−5=0$

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4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

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y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

Matrix

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$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $