Solve 17e^2+18e+19=0

$$17 e ^ { 2 } + 18 e + 19 = 0$$

Verify

Steps Using Direct Factoring Method

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 17

$$x ^ 2 +\frac{18}{17}x +\frac{19}{17} = 0$$

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

$$r + s = -\frac{18}{17} rs = \frac{19}{17}$$

Two numbers r and s sum up to -\frac{18}{17} exactly when the average of the two numbers is \frac{1}{2}*-\frac{18}{17} = -\frac{9}{17}. 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>

$$r = -\frac{9}{17} - u s = -\frac{9}{17} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{19}{17}

$$(-\frac{9}{17} - u) (-\frac{9}{17} + u) = \frac{19}{17}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{81}{289} - u^2 = \frac{19}{17}$$

Simplify the expression by subtracting \frac{81}{289} on both sides

$$-u^2 = \frac{19}{17}-\frac{81}{289} = \frac{242}{289}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = -\frac{242}{289} u = \pm\sqrt{-\frac{242}{289}} = \pm \frac{\sqrt{242}}{17}i$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{9}{17} - \frac{\sqrt{242}}{17}i = -0.529 - 0.915i s = -\frac{9}{17} + \frac{\sqrt{242}}{17}i = -0.529 + 0.915i$$