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417x^{2}+13x-2=0
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{-13±\sqrt{13^{2}-4\times 417\left(-2\right)}}{2\times 417}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 417 for a, 13 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-13±\sqrt{169-4\times 417\left(-2\right)}}{2\times 417}
Square 13.
x=\frac{-13±\sqrt{169-1668\left(-2\right)}}{2\times 417}
Multiply -4 times 417.
x=\frac{-13±\sqrt{169+3336}}{2\times 417}
Multiply -1668 times -2.
x=\frac{-13±\sqrt{3505}}{2\times 417}
Add 169 to 3336.
x=\frac{-13±\sqrt{3505}}{834}
Multiply 2 times 417.
x=\frac{\sqrt{3505}-13}{834}
Now solve the equation x=\frac{-13±\sqrt{3505}}{834} when ± is plus. Add -13 to \sqrt{3505}.
x=\frac{-\sqrt{3505}-13}{834}
Now solve the equation x=\frac{-13±\sqrt{3505}}{834} when ± is minus. Subtract \sqrt{3505} from -13.
x=\frac{\sqrt{3505}-13}{834} x=\frac{-\sqrt{3505}-13}{834}
The equation is now solved.
417x^{2}+13x-2=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
417x^{2}+13x-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
417x^{2}+13x=-\left(-2\right)
Subtracting -2 from itself leaves 0.
417x^{2}+13x=2
Subtract -2 from 0.
\frac{417x^{2}+13x}{417}=\frac{2}{417}
Divide both sides by 417.
x^{2}+\frac{13}{417}x=\frac{2}{417}
Dividing by 417 undoes the multiplication by 417.
x^{2}+\frac{13}{417}x+\left(\frac{13}{834}\right)^{2}=\frac{2}{417}+\left(\frac{13}{834}\right)^{2}
Divide \frac{13}{417}, the coefficient of the x term, by 2 to get \frac{13}{834}. Then add the square of \frac{13}{834} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{13}{417}x+\frac{169}{695556}=\frac{2}{417}+\frac{169}{695556}
Square \frac{13}{834} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{13}{417}x+\frac{169}{695556}=\frac{3505}{695556}
Add \frac{2}{417} to \frac{169}{695556} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{13}{834}\right)^{2}=\frac{3505}{695556}
Factor x^{2}+\frac{13}{417}x+\frac{169}{695556}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{13}{834}\right)^{2}}=\sqrt{\frac{3505}{695556}}
Take the square root of both sides of the equation.
x+\frac{13}{834}=\frac{\sqrt{3505}}{834} x+\frac{13}{834}=-\frac{\sqrt{3505}}{834}
Simplify.
x=\frac{\sqrt{3505}-13}{834} x=\frac{-\sqrt{3505}-13}{834}
Subtract \frac{13}{834} from both sides of the equation.
x ^ 2 +\frac{13}{417}x -\frac{2}{417} = 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 417
r + s = -\frac{13}{417} rs = -\frac{2}{417}
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{13}{834} - u s = -\frac{13}{834} + u
Two numbers r and s sum up to -\frac{13}{417} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{417} = -\frac{13}{834}. 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{13}{834} - u) (-\frac{13}{834} + u) = -\frac{2}{417}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{417}
\frac{169}{695556} - u^2 = -\frac{2}{417}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{2}{417}-\frac{169}{695556} = \frac{3505}{695556}
Simplify the expression by subtracting \frac{169}{695556} on both sides
u^2 = -\frac{3505}{695556} u = \pm\sqrt{-\frac{3505}{695556}} = \pm \frac{\sqrt{3505}}{834}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{13}{834} - \frac{\sqrt{3505}}{834}i = -0.087 s = -\frac{13}{834} + \frac{\sqrt{3505}}{834}i = 0.055
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.