Solve n^2-19n+81

$$n ^ { 2 } - 19 n + 81$$

Factor

$$\left(n-\frac{19-\sqrt{37}}{2}\right)\left(n-\frac{\sqrt{37}+19}{2}\right)$$

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.

$$n^{2}-19n+81=0$$

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.

$$n=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 81}}{2}$$

Square -19.

$$n=\frac{-\left(-19\right)±\sqrt{361-4\times 81}}{2}$$

Multiply -4 times 81.

$$n=\frac{-\left(-19\right)±\sqrt{361-324}}{2}$$

Add 361 to -324.

$$n=\frac{-\left(-19\right)±\sqrt{37}}{2}$$

The opposite of -19 is 19.

$$n=\frac{19±\sqrt{37}}{2}$$

Now solve the equation n=\frac{19±\sqrt{37}}{2} when ± is plus. Add 19 to \sqrt{37}.

$$n=\frac{\sqrt{37}+19}{2}$$

Now solve the equation n=\frac{19±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from 19.

$$n=\frac{19-\sqrt{37}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{37}}{2} for x_{1} and \frac{19-\sqrt{37}}{2} for x_{2}.

$$n^{2}-19n+81=\left(n-\frac{\sqrt{37}+19}{2}\right)\left(n-\frac{19-\sqrt{37}}{2}\right)$$

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.

$$x ^ 2 -19x +81 = 0$$

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

$$r + s = 19 rs = 81$$

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

$$r = \frac{19}{2} - u s = \frac{19}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 81

$$(\frac{19}{2} - u) (\frac{19}{2} + u) = 81$$

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

$$\frac{361}{4} - u^2 = 81$$

Simplify the expression by subtracting \frac{361}{4} on both sides

$$-u^2 = 81-\frac{361}{4} = -\frac{37}{4}$$

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

$$u^2 = \frac{37}{4} u = \pm\sqrt{\frac{37}{4}} = \pm \frac{\sqrt{37}}{2}$$

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

$$r =\frac{19}{2} - \frac{\sqrt{37}}{2} = 6.459 s = \frac{19}{2} + \frac{\sqrt{37}}{2} = 12.541$$

Evaluate

$$n^{2}-19n+81$$