Solve 72n^2-76n-8

$$72 n ^ { 2 } - 76 n - 8$$

Factor

$$72\left(n-\frac{19-\sqrt{505}}{36}\right)\left(n-\frac{\sqrt{505}+19}{36}\right)$$

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.

$$72n^{2}-76n-8=0$$

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.

$$n=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 72\left(-8\right)}}{2\times 72}$$

Square -76.

$$n=\frac{-\left(-76\right)±\sqrt{5776-4\times 72\left(-8\right)}}{2\times 72}$$

Multiply -4 times 72.

$$n=\frac{-\left(-76\right)±\sqrt{5776-288\left(-8\right)}}{2\times 72}$$

Multiply -288 times -8.

$$n=\frac{-\left(-76\right)±\sqrt{5776+2304}}{2\times 72}$$

Add 5776 to 2304.

$$n=\frac{-\left(-76\right)±\sqrt{8080}}{2\times 72}$$

Take the square root of 8080.

$$n=\frac{-\left(-76\right)±4\sqrt{505}}{2\times 72}$$

The opposite of -76 is 76.

$$n=\frac{76±4\sqrt{505}}{2\times 72}$$

Multiply 2 times 72.

$$n=\frac{76±4\sqrt{505}}{144}$$

Now solve the equation n=\frac{76±4\sqrt{505}}{144} when ± is plus. Add 76 to 4\sqrt{505}.

$$n=\frac{4\sqrt{505}+76}{144}$$

Divide 76+4\sqrt{505} by 144.

$$n=\frac{\sqrt{505}+19}{36}$$

Now solve the equation n=\frac{76±4\sqrt{505}}{144} when ± is minus. Subtract 4\sqrt{505} from 76.

$$n=\frac{76-4\sqrt{505}}{144}$$

Divide 76-4\sqrt{505} by 144.

$$n=\frac{19-\sqrt{505}}{36}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{505}}{36} for x_{1} and \frac{19-\sqrt{505}}{36} for x_{2}.

$$72n^{2}-76n-8=72\left(n-\frac{\sqrt{505}+19}{36}\right)\left(n-\frac{19-\sqrt{505}}{36}\right)$$

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 72

$$x ^ 2 -\frac{19}{18}x -\frac{1}{9} = 0$$

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

$$r + s = \frac{19}{18} rs = -\frac{1}{9}$$

Two numbers r and s sum up to \frac{19}{18} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{18} = \frac{19}{36}. 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>

$$r = \frac{19}{36} - u s = \frac{19}{36} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{9}

$$(\frac{19}{36} - u) (\frac{19}{36} + u) = -\frac{1}{9}$$

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

$$\frac{361}{1296} - u^2 = -\frac{1}{9}$$

Simplify the expression by subtracting \frac{361}{1296} on both sides

$$-u^2 = -\frac{1}{9}-\frac{361}{1296} = -\frac{505}{1296}$$

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

$$u^2 = \frac{505}{1296} u = \pm\sqrt{\frac{505}{1296}} = \pm \frac{\sqrt{505}}{36}$$

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

$$r =\frac{19}{36} - \frac{\sqrt{505}}{36} = -0.096 s = \frac{19}{36} + \frac{\sqrt{505}}{36} = 1.152$$

Evaluate

$$72n^{2}-76n-8$$