Factor
3\left(u-\left(-\frac{\sqrt{66}}{3}-3\right)\right)\left(u-\left(\frac{\sqrt{66}}{3}-3\right)\right)
Evaluate
3u^{2}+18u+5
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3u^{2}+18u+5=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.
u=\frac{-18±\sqrt{18^{2}-4\times 3\times 5}}{2\times 3}
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.
u=\frac{-18±\sqrt{324-4\times 3\times 5}}{2\times 3}
Square 18.
u=\frac{-18±\sqrt{324-12\times 5}}{2\times 3}
Multiply -4 times 3.
u=\frac{-18±\sqrt{324-60}}{2\times 3}
Multiply -12 times 5.
u=\frac{-18±\sqrt{264}}{2\times 3}
Add 324 to -60.
u=\frac{-18±2\sqrt{66}}{2\times 3}
Take the square root of 264.
u=\frac{-18±2\sqrt{66}}{6}
Multiply 2 times 3.
u=\frac{2\sqrt{66}-18}{6}
Now solve the equation u=\frac{-18±2\sqrt{66}}{6} when ± is plus. Add -18 to 2\sqrt{66}.
u=\frac{\sqrt{66}}{3}-3
Divide -18+2\sqrt{66} by 6.
u=\frac{-2\sqrt{66}-18}{6}
Now solve the equation u=\frac{-18±2\sqrt{66}}{6} when ± is minus. Subtract 2\sqrt{66} from -18.
u=-\frac{\sqrt{66}}{3}-3
Divide -18-2\sqrt{66} by 6.
3u^{2}+18u+5=3\left(u-\left(\frac{\sqrt{66}}{3}-3\right)\right)\left(u-\left(-\frac{\sqrt{66}}{3}-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3+\frac{\sqrt{66}}{3} for x_{1} and -3-\frac{\sqrt{66}}{3} for x_{2}.
x ^ 2 +6x +\frac{5}{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 = -6 rs = \frac{5}{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 = -3 - u s = -3 + u
Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -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>
(-3 - u) (-3 + u) = \frac{5}{3}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{3}
9 - u^2 = \frac{5}{3}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{5}{3}-9 = -\frac{22}{3}
Simplify the expression by subtracting 9 on both sides
u^2 = \frac{22}{3} u = \pm\sqrt{\frac{22}{3}} = \pm \frac{\sqrt{22}}{\sqrt{3}}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-3 - \frac{\sqrt{22}}{\sqrt{3}} = -5.708 s = -3 + \frac{\sqrt{22}}{\sqrt{3}} = -0.292
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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}