Factor
\left(x-\left(19-\sqrt{353}\right)\right)\left(x-\left(\sqrt{353}+19\right)\right)
Evaluate
x^{2}-38x+8
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x^{2}-38x+8=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{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 8}}{2}
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{-\left(-38\right)±\sqrt{1444-4\times 8}}{2}
Square -38.
x=\frac{-\left(-38\right)±\sqrt{1444-32}}{2}
Multiply -4 times 8.
x=\frac{-\left(-38\right)±\sqrt{1412}}{2}
Add 1444 to -32.
x=\frac{-\left(-38\right)±2\sqrt{353}}{2}
Take the square root of 1412.
x=\frac{38±2\sqrt{353}}{2}
The opposite of -38 is 38.
x=\frac{2\sqrt{353}+38}{2}
Now solve the equation x=\frac{38±2\sqrt{353}}{2} when ± is plus. Add 38 to 2\sqrt{353}.
x=\sqrt{353}+19
Divide 38+2\sqrt{353} by 2.
x=\frac{38-2\sqrt{353}}{2}
Now solve the equation x=\frac{38±2\sqrt{353}}{2} when ± is minus. Subtract 2\sqrt{353} from 38.
x=19-\sqrt{353}
Divide 38-2\sqrt{353} by 2.
x^{2}-38x+8=\left(x-\left(\sqrt{353}+19\right)\right)\left(x-\left(19-\sqrt{353}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 19+\sqrt{353} for x_{1} and 19-\sqrt{353} for x_{2}.
x ^ 2 -38x +8 = 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.
r + s = 38 rs = 8
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 = 19 - u s = 19 + u
Two numbers r and s sum up to 38 exactly when the average of the two numbers is \frac{1}{2}*38 = 19. 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>
(19 - u) (19 + u) = 8
To solve for unknown quantity u, substitute these in the product equation rs = 8
361 - u^2 = 8
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 8-361 = -353
Simplify the expression by subtracting 361 on both sides
u^2 = 353 u = \pm\sqrt{353} = \pm \sqrt{353}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =19 - \sqrt{353} = 0.212 s = 19 + \sqrt{353} = 37.788
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 } - 4 x - 5 = 0
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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}