Solve 6n^2+54n-360

$$6 n ^ { 2 } + 54 n - 360$$

Factor

$$6\left(n-\frac{-\sqrt{321}-9}{2}\right)\left(n-\frac{\sqrt{321}-9}{2}\right)$$

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.

$$6n^{2}+54n-360=0$$

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.

$$n=\frac{-54±\sqrt{54^{2}-4\times 6\left(-360\right)}}{2\times 6}$$

Square 54.

$$n=\frac{-54±\sqrt{2916-4\times 6\left(-360\right)}}{2\times 6}$$

Multiply -4 times 6.

$$n=\frac{-54±\sqrt{2916-24\left(-360\right)}}{2\times 6}$$

Multiply -24 times -360.

$$n=\frac{-54±\sqrt{2916+8640}}{2\times 6}$$

Add 2916 to 8640.

$$n=\frac{-54±\sqrt{11556}}{2\times 6}$$

Take the square root of 11556.

$$n=\frac{-54±6\sqrt{321}}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{-54±6\sqrt{321}}{12}$$

Now solve the equation n=\frac{-54±6\sqrt{321}}{12} when ± is plus. Add -54 to 6\sqrt{321}.

$$n=\frac{6\sqrt{321}-54}{12}$$

Divide -54+6\sqrt{321} by 12.

$$n=\frac{\sqrt{321}-9}{2}$$

Now solve the equation n=\frac{-54±6\sqrt{321}}{12} when ± is minus. Subtract 6\sqrt{321} from -54.

$$n=\frac{-6\sqrt{321}-54}{12}$$

Divide -54-6\sqrt{321} by 12.

$$n=\frac{-\sqrt{321}-9}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+\sqrt{321}}{2} for x_{1} and \frac{-9-\sqrt{321}}{2} for x_{2}.

$$6n^{2}+54n-360=6\left(n-\frac{\sqrt{321}-9}{2}\right)\left(n-\frac{-\sqrt{321}-9}{2}\right)$$

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.This is achieved by dividing both sides of the equation by 6

$$x ^ 2 +9x -60 = 0$$

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

$$r + s = -9 rs = -60$$

Two numbers r and s sum up to -9 exactly when the average of the two numbers is \frac{1}{2}*-9 = -\frac{9}{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>

$$r = -\frac{9}{2} - u s = -\frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -60

$$(-\frac{9}{2} - u) (-\frac{9}{2} + u) = -60$$

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

$$\frac{81}{4} - u^2 = -60$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = -60-\frac{81}{4} = -\frac{321}{4}$$

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

$$u^2 = \frac{321}{4} u = \pm\sqrt{\frac{321}{4}} = \pm \frac{\sqrt{321}}{2}$$

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

$$r =-\frac{9}{2} - \frac{\sqrt{321}}{2} = -13.458 s = -\frac{9}{2} + \frac{\sqrt{321}}{2} = 4.458$$

Evaluate

$$6\left(n^{2}+9n-60\right)$$