Solve 2e^2+7e+5

$$2 e ^ { 2 } + 7 e + 5$$

Evaluate

$$2e^{2}+7e+5\approx 38.806084997$$

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 2

$$x ^ 2 +\frac{7}{2}x +\frac{5}{2} = 0$$

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

$$r + s = -\frac{7}{2} rs = \frac{5}{2}$$

Two numbers r and s sum up to -\frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{2} = -\frac{7}{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>

$$r = -\frac{7}{4} - u s = -\frac{7}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{2}

$$(-\frac{7}{4} - u) (-\frac{7}{4} + u) = \frac{5}{2}$$

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

$$\frac{49}{16} - u^2 = \frac{5}{2}$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = \frac{5}{2}-\frac{49}{16} = -\frac{9}{16}$$

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

$$u^2 = \frac{9}{16} u = \pm\sqrt{\frac{9}{16}} = \pm \frac{3}{4}$$

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

$$r =-\frac{7}{4} - \frac{3}{4} = -2.500 s = -\frac{7}{4} + \frac{3}{4} = -1$$