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5x^{2}-40x-4=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(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 5\left(-4\right)}}{2\times 5}
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(-40\right)±\sqrt{1600-4\times 5\left(-4\right)}}{2\times 5}
Square -40.
x=\frac{-\left(-40\right)±\sqrt{1600-20\left(-4\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-\left(-40\right)±\sqrt{1600+80}}{2\times 5}
Multiply -20 times -4.
x=\frac{-\left(-40\right)±\sqrt{1680}}{2\times 5}
Add 1600 to 80.
x=\frac{-\left(-40\right)±4\sqrt{105}}{2\times 5}
Take the square root of 1680.
x=\frac{40±4\sqrt{105}}{2\times 5}
The opposite of -40 is 40.
x=\frac{40±4\sqrt{105}}{10}
Multiply 2 times 5.
x=\frac{4\sqrt{105}+40}{10}
Now solve the equation x=\frac{40±4\sqrt{105}}{10} when ± is plus. Add 40 to 4\sqrt{105}.
x=\frac{2\sqrt{105}}{5}+4
Divide 40+4\sqrt{105} by 10.
x=\frac{40-4\sqrt{105}}{10}
Now solve the equation x=\frac{40±4\sqrt{105}}{10} when ± is minus. Subtract 4\sqrt{105} from 40.
x=-\frac{2\sqrt{105}}{5}+4
Divide 40-4\sqrt{105} by 10.
5x^{2}-40x-4=5\left(x-\left(\frac{2\sqrt{105}}{5}+4\right)\right)\left(x-\left(-\frac{2\sqrt{105}}{5}+4\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4+\frac{2\sqrt{105}}{5} for x_{1} and 4-\frac{2\sqrt{105}}{5} for x_{2}.
x ^ 2 -8x -\frac{4}{5} = 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 5
r + s = 8 rs = -\frac{4}{5}
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 = 4 - u s = 4 + u
Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. 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>
(4 - u) (4 + u) = -\frac{4}{5}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{5}
16 - u^2 = -\frac{4}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{5}-16 = -\frac{84}{5}
Simplify the expression by subtracting 16 on both sides
u^2 = \frac{84}{5} u = \pm\sqrt{\frac{84}{5}} = \pm \frac{\sqrt{84}}{\sqrt{5}}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =4 - \frac{\sqrt{84}}{\sqrt{5}} = -0.099 s = 4 + \frac{\sqrt{84}}{\sqrt{5}} = 8.099
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.