Solve 11n^2+105n+2

$$11 n ^ { 2 } + 105 n + 2$$

Factor

$$11\left(n-\frac{-\sqrt{10937}-105}{22}\right)\left(n-\frac{\sqrt{10937}-105}{22}\right)$$

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.

$$11n^{2}+105n+2=0$$

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.

$$n=\frac{-105±\sqrt{105^{2}-4\times 11\times 2}}{2\times 11}$$

Square 105.

$$n=\frac{-105±\sqrt{11025-4\times 11\times 2}}{2\times 11}$$

Multiply -4 times 11.

$$n=\frac{-105±\sqrt{11025-44\times 2}}{2\times 11}$$

Multiply -44 times 2.

$$n=\frac{-105±\sqrt{11025-88}}{2\times 11}$$

Add 11025 to -88.

$$n=\frac{-105±\sqrt{10937}}{2\times 11}$$

Multiply 2 times 11.

$$n=\frac{-105±\sqrt{10937}}{22}$$

Now solve the equation n=\frac{-105±\sqrt{10937}}{22} when ± is plus. Add -105 to \sqrt{10937}.

$$n=\frac{\sqrt{10937}-105}{22}$$

Now solve the equation n=\frac{-105±\sqrt{10937}}{22} when ± is minus. Subtract \sqrt{10937} from -105.

$$n=\frac{-\sqrt{10937}-105}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-105+\sqrt{10937}}{22} for x_{1} and \frac{-105-\sqrt{10937}}{22} for x_{2}.

$$11n^{2}+105n+2=11\left(n-\frac{\sqrt{10937}-105}{22}\right)\left(n-\frac{-\sqrt{10937}-105}{22}\right)$$

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 11

$$x ^ 2 +\frac{105}{11}x +\frac{2}{11} = 0$$

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

$$r + s = -\frac{105}{11} rs = \frac{2}{11}$$

Two numbers r and s sum up to -\frac{105}{11} exactly when the average of the two numbers is \frac{1}{2}*-\frac{105}{11} = -\frac{105}{22}. 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>

$$r = -\frac{105}{22} - u s = -\frac{105}{22} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2}{11}

$$(-\frac{105}{22} - u) (-\frac{105}{22} + u) = \frac{2}{11}$$

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

$$\frac{11025}{484} - u^2 = \frac{2}{11}$$

Simplify the expression by subtracting \frac{11025}{484} on both sides

$$-u^2 = \frac{2}{11}-\frac{11025}{484} = -\frac{10937}{484}$$

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

$$u^2 = \frac{10937}{484} u = \pm\sqrt{\frac{10937}{484}} = \pm \frac{\sqrt{10937}}{22}$$

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

$$r =-\frac{105}{22} - \frac{\sqrt{10937}}{22} = -9.526 s = -\frac{105}{22} + \frac{\sqrt{10937}}{22} = -0.019$$

Evaluate

$$11n^{2}+105n+2$$