Factor
4\left(a-\frac{1-\sqrt{2}}{2}\right)\left(a-\frac{\sqrt{2}+1}{2}\right)
Evaluate
4a^{2}-4a-1
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4a^{2}-4a-1=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.
a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}
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.
a=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-1\right)}}{2\times 4}
Square -4.
a=\frac{-\left(-4\right)±\sqrt{16-16\left(-1\right)}}{2\times 4}
Multiply -4 times 4.
a=\frac{-\left(-4\right)±\sqrt{16+16}}{2\times 4}
Multiply -16 times -1.
a=\frac{-\left(-4\right)±\sqrt{32}}{2\times 4}
Add 16 to 16.
a=\frac{-\left(-4\right)±4\sqrt{2}}{2\times 4}
Take the square root of 32.
a=\frac{4±4\sqrt{2}}{2\times 4}
The opposite of -4 is 4.
a=\frac{4±4\sqrt{2}}{8}
Multiply 2 times 4.
a=\frac{4\sqrt{2}+4}{8}
Now solve the equation a=\frac{4±4\sqrt{2}}{8} when ± is plus. Add 4 to 4\sqrt{2}.
a=\frac{\sqrt{2}+1}{2}
Divide 4+4\sqrt{2} by 8.
a=\frac{4-4\sqrt{2}}{8}
Now solve the equation a=\frac{4±4\sqrt{2}}{8} when ± is minus. Subtract 4\sqrt{2} from 4.
a=\frac{1-\sqrt{2}}{2}
Divide 4-4\sqrt{2} by 8.
4a^{2}-4a-1=4\left(a-\frac{\sqrt{2}+1}{2}\right)\left(a-\frac{1-\sqrt{2}}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{2}}{2} for x_{1} and \frac{1-\sqrt{2}}{2} for x_{2}.
x ^ 2 -1x -\frac{1}{4} = 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 4
r + s = 1 rs = -\frac{1}{4}
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}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{2}. 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}{2} - u) (\frac{1}{2} + u) = -\frac{1}{4}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{4}
\frac{1}{4} - u^2 = -\frac{1}{4}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{1}{4}-\frac{1}{4} = -\frac{1}{2}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{1}{2} u = \pm\sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}
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}{2} - \frac{1}{\sqrt{2}} = -0.207 s = \frac{1}{2} + \frac{1}{\sqrt{2}} = 1.207
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}