Solve n^2-14n-77

$$n ^ { 2 } - 14 n - 77$$

Factor

$$\left(n-\left(7-3\sqrt{14}\right)\right)\left(n-\left(3\sqrt{14}+7\right)\right)$$

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.

$$n^{2}-14n-77=0$$

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.

$$n=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-77\right)}}{2}$$

Square -14.

$$n=\frac{-\left(-14\right)±\sqrt{196-4\left(-77\right)}}{2}$$

Multiply -4 times -77.

$$n=\frac{-\left(-14\right)±\sqrt{196+308}}{2}$$

Add 196 to 308.

$$n=\frac{-\left(-14\right)±\sqrt{504}}{2}$$

Take the square root of 504.

$$n=\frac{-\left(-14\right)±6\sqrt{14}}{2}$$

The opposite of -14 is 14.

$$n=\frac{14±6\sqrt{14}}{2}$$

Now solve the equation n=\frac{14±6\sqrt{14}}{2} when ± is plus. Add 14 to 6\sqrt{14}.

$$n=\frac{6\sqrt{14}+14}{2}$$

Divide 14+6\sqrt{14} by 2.

$$n=3\sqrt{14}+7$$

Now solve the equation n=\frac{14±6\sqrt{14}}{2} when ± is minus. Subtract 6\sqrt{14} from 14.

$$n=\frac{14-6\sqrt{14}}{2}$$

Divide 14-6\sqrt{14} by 2.

$$n=7-3\sqrt{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7+3\sqrt{14} for x_{1} and 7-3\sqrt{14} for x_{2}.

$$n^{2}-14n-77=\left(n-\left(3\sqrt{14}+7\right)\right)\left(n-\left(7-3\sqrt{14}\right)\right)$$

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.

$$x ^ 2 -14x -77 = 0$$

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

$$r + s = 14 rs = -77$$

Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>

$$r = 7 - u s = 7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -77

$$(7 - u) (7 + u) = -77$$

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

$$49 - u^2 = -77$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = -77-49 = -126$$

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

$$u^2 = 126 u = \pm\sqrt{126} = \pm \sqrt{126}$$

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

$$r =7 - \sqrt{126} = -4.225 s = 7 + \sqrt{126} = 18.225$$

Evaluate

$$n^{2}-14n-77$$