Solve 2x^2+36x+17

$$2 x ^ { 2 } + 36 x + 17$$

Factor

$$2\left(x-\left(-\frac{\sqrt{290}}{2}-9\right)\right)\left(x-\left(\frac{\sqrt{290}}{2}-9\right)\right)$$

Steps Using the Quadratic Formula

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.

$$2x^{2}+36x+17=0$$

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.

$$x=\frac{-36±\sqrt{36^{2}-4\times 2\times 17}}{2\times 2}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 2\times 17}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-36±\sqrt{1296-8\times 17}}{2\times 2}$$

Multiply -8 times 17.

$$x=\frac{-36±\sqrt{1296-136}}{2\times 2}$$

Add 1296 to -136.

$$x=\frac{-36±\sqrt{1160}}{2\times 2}$$

Take the square root of 1160.

$$x=\frac{-36±2\sqrt{290}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-36±2\sqrt{290}}{4}$$

Now solve the equation x=\frac{-36±2\sqrt{290}}{4} when ± is plus. Add -36 to 2\sqrt{290}.

$$x=\frac{2\sqrt{290}-36}{4}$$

Divide -36+2\sqrt{290} by 4.

$$x=\frac{\sqrt{290}}{2}-9$$

Now solve the equation x=\frac{-36±2\sqrt{290}}{4} when ± is minus. Subtract 2\sqrt{290} from -36.

$$x=\frac{-2\sqrt{290}-36}{4}$$

Divide -36-2\sqrt{290} by 4.

$$x=-\frac{\sqrt{290}}{2}-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -9+\frac{\sqrt{290}}{2} for x_{1} and -9-\frac{\sqrt{290}}{2} for x_{2}.

$$2x^{2}+36x+17=2\left(x-\left(\frac{\sqrt{290}}{2}-9\right)\right)\left(x-\left(-\frac{\sqrt{290}}{2}-9\right)\right)$$

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

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

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

$$r + s = -18 rs = \frac{17}{2}$$

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

$$r = -9 - u s = -9 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{17}{2}

$$(-9 - u) (-9 + u) = \frac{17}{2}$$

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

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

Simplify the expression by subtracting 81 on both sides

$$-u^2 = \frac{17}{2}-81 = -\frac{145}{2}$$

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

$$u^2 = \frac{145}{2} u = \pm\sqrt{\frac{145}{2}} = \pm \frac{\sqrt{145}}{\sqrt{2}}$$

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

$$r =-9 - \frac{\sqrt{145}}{\sqrt{2}} = -17.515 s = -9 + \frac{\sqrt{145}}{\sqrt{2}} = -0.485$$

Evaluate

$$2x^{2}+36x+17$$