Solve n^2-36n-72Rightarrow0

$$n ^ { 2 } - 36 n - 72 \Rightarrow 0$$

Factor

$$\left(n-\left(18-6\sqrt{11}\right)\right)\left(n-\left(6\sqrt{11}+18\right)\right)$$

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.

$$n^{2}-36n-72=0$$

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.

$$n=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\left(-72\right)}}{2}$$

Square -36.

$$n=\frac{-\left(-36\right)±\sqrt{1296-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$n=\frac{-\left(-36\right)±\sqrt{1296+288}}{2}$$

Add 1296 to 288.

$$n=\frac{-\left(-36\right)±\sqrt{1584}}{2}$$

Take the square root of 1584.

$$n=\frac{-\left(-36\right)±12\sqrt{11}}{2}$$

The opposite of -36 is 36.

$$n=\frac{36±12\sqrt{11}}{2}$$

Now solve the equation n=\frac{36±12\sqrt{11}}{2} when ± is plus. Add 36 to 12\sqrt{11}.

$$n=\frac{12\sqrt{11}+36}{2}$$

Divide 36+12\sqrt{11} by 2.

$$n=6\sqrt{11}+18$$

Now solve the equation n=\frac{36±12\sqrt{11}}{2} when ± is minus. Subtract 12\sqrt{11} from 36.

$$n=\frac{36-12\sqrt{11}}{2}$$

Divide 36-12\sqrt{11} by 2.

$$n=18-6\sqrt{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 18+6\sqrt{11} for x_{1} and 18-6\sqrt{11} for x_{2}.

$$n^{2}-36n-72=\left(n-\left(6\sqrt{11}+18\right)\right)\left(n-\left(18-6\sqrt{11}\right)\right)$$

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.

$$x ^ 2 -36x -72 = 0$$

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

$$r + s = 36 rs = -72$$

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

$$r = 18 - u s = 18 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -72

$$(18 - u) (18 + u) = -72$$

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

$$324 - u^2 = -72$$

Simplify the expression by subtracting 324 on both sides

$$-u^2 = -72-324 = -396$$

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

$$u^2 = 396 u = \pm\sqrt{396} = \pm \sqrt{396}$$

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

$$r =18 - \sqrt{396} = -1.900 s = 18 + \sqrt{396} = 37.900$$

Evaluate

$$n^{2}-36n-72$$