Factor
\left(x-\left(12-\sqrt{141}\right)\right)\left(x-\left(\sqrt{141}+12\right)\right)
Evaluate
x^{2}-24x+3
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x^{2}-24x+3=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(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 3}}{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(-24\right)±\sqrt{576-4\times 3}}{2}
Square -24.
x=\frac{-\left(-24\right)±\sqrt{576-12}}{2}
Multiply -4 times 3.
x=\frac{-\left(-24\right)±\sqrt{564}}{2}
Add 576 to -12.
x=\frac{-\left(-24\right)±2\sqrt{141}}{2}
Take the square root of 564.
x=\frac{24±2\sqrt{141}}{2}
The opposite of -24 is 24.
x=\frac{2\sqrt{141}+24}{2}
Now solve the equation x=\frac{24±2\sqrt{141}}{2} when ± is plus. Add 24 to 2\sqrt{141}.
x=\sqrt{141}+12
Divide 24+2\sqrt{141} by 2.
x=\frac{24-2\sqrt{141}}{2}
Now solve the equation x=\frac{24±2\sqrt{141}}{2} when ± is minus. Subtract 2\sqrt{141} from 24.
x=12-\sqrt{141}
Divide 24-2\sqrt{141} by 2.
x^{2}-24x+3=\left(x-\left(\sqrt{141}+12\right)\right)\left(x-\left(12-\sqrt{141}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+\sqrt{141} for x_{1} and 12-\sqrt{141} for x_{2}.
x ^ 2 -24x +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.
r + s = 24 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 = 12 - u s = 12 + u
Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 12. 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>
(12 - u) (12 + u) = 3
To solve for unknown quantity u, substitute these in the product equation rs = 3
144 - u^2 = 3
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 3-144 = -141
Simplify the expression by subtracting 144 on both sides
u^2 = 141 u = \pm\sqrt{141} = \pm \sqrt{141}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =12 - \sqrt{141} = 0.126 s = 12 + \sqrt{141} = 23.874
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}