Solve 8n^2+116n+401

$$8 n ^ { 2 } + 116 n + 401$$

Factor

$$8\left(n-\frac{-\sqrt{39}-29}{4}\right)\left(n-\frac{\sqrt{39}-29}{4}\right)$$

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.

$$8n^{2}+116n+401=0$$

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.

$$n=\frac{-116±\sqrt{116^{2}-4\times 8\times 401}}{2\times 8}$$

Square 116.

$$n=\frac{-116±\sqrt{13456-4\times 8\times 401}}{2\times 8}$$

Multiply -4 times 8.

$$n=\frac{-116±\sqrt{13456-32\times 401}}{2\times 8}$$

Multiply -32 times 401.

$$n=\frac{-116±\sqrt{13456-12832}}{2\times 8}$$

Add 13456 to -12832.

$$n=\frac{-116±\sqrt{624}}{2\times 8}$$

Take the square root of 624.

$$n=\frac{-116±4\sqrt{39}}{2\times 8}$$

Multiply 2 times 8.

$$n=\frac{-116±4\sqrt{39}}{16}$$

Now solve the equation n=\frac{-116±4\sqrt{39}}{16} when ± is plus. Add -116 to 4\sqrt{39}.

$$n=\frac{4\sqrt{39}-116}{16}$$

Divide -116+4\sqrt{39} by 16.

$$n=\frac{\sqrt{39}-29}{4}$$

Now solve the equation n=\frac{-116±4\sqrt{39}}{16} when ± is minus. Subtract 4\sqrt{39} from -116.

$$n=\frac{-4\sqrt{39}-116}{16}$$

Divide -116-4\sqrt{39} by 16.

$$n=\frac{-\sqrt{39}-29}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-29+\sqrt{39}}{4} for x_{1} and \frac{-29-\sqrt{39}}{4} for x_{2}.

$$8n^{2}+116n+401=8\left(n-\frac{\sqrt{39}-29}{4}\right)\left(n-\frac{-\sqrt{39}-29}{4}\right)$$

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 8

$$x ^ 2 +\frac{29}{2}x +\frac{401}{8} = 0$$

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

$$r + s = -\frac{29}{2} rs = \frac{401}{8}$$

Two numbers r and s sum up to -\frac{29}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{29}{2} = -\frac{29}{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>

$$r = -\frac{29}{4} - u s = -\frac{29}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{401}{8}

$$(-\frac{29}{4} - u) (-\frac{29}{4} + u) = \frac{401}{8}$$

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

$$\frac{841}{16} - u^2 = \frac{401}{8}$$

Simplify the expression by subtracting \frac{841}{16} on both sides

$$-u^2 = \frac{401}{8}-\frac{841}{16} = -\frac{39}{16}$$

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

$$u^2 = \frac{39}{16} u = \pm\sqrt{\frac{39}{16}} = \pm \frac{\sqrt{39}}{4}$$

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

$$r =-\frac{29}{4} - \frac{\sqrt{39}}{4} = -8.811 s = -\frac{29}{4} + \frac{\sqrt{39}}{4} = -5.689$$

Evaluate

$$8n^{2}+116n+401$$