Solve 3n^2-359n+10620

$$3 n ^ { 2 } - 359 n + 10620$$

Factor

$$3\left(n-\frac{359-\sqrt{1441}}{6}\right)\left(n-\frac{\sqrt{1441}+359}{6}\right)$$

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.

$$3n^{2}-359n+10620=0$$

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.

$$n=\frac{-\left(-359\right)±\sqrt{\left(-359\right)^{2}-4\times 3\times 10620}}{2\times 3}$$

Square -359.

$$n=\frac{-\left(-359\right)±\sqrt{128881-4\times 3\times 10620}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-359\right)±\sqrt{128881-12\times 10620}}{2\times 3}$$

Multiply -12 times 10620.

$$n=\frac{-\left(-359\right)±\sqrt{128881-127440}}{2\times 3}$$

Add 128881 to -127440.

$$n=\frac{-\left(-359\right)±\sqrt{1441}}{2\times 3}$$

The opposite of -359 is 359.

$$n=\frac{359±\sqrt{1441}}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{359±\sqrt{1441}}{6}$$

Now solve the equation n=\frac{359±\sqrt{1441}}{6} when ± is plus. Add 359 to \sqrt{1441}.

$$n=\frac{\sqrt{1441}+359}{6}$$

Now solve the equation n=\frac{359±\sqrt{1441}}{6} when ± is minus. Subtract \sqrt{1441} from 359.

$$n=\frac{359-\sqrt{1441}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{359+\sqrt{1441}}{6} for x_{1} and \frac{359-\sqrt{1441}}{6} for x_{2}.

$$3n^{2}-359n+10620=3\left(n-\frac{\sqrt{1441}+359}{6}\right)\left(n-\frac{359-\sqrt{1441}}{6}\right)$$

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 3

$$x ^ 2 -\frac{359}{3}x +3540 = 0$$

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

$$r + s = \frac{359}{3} rs = 3540$$

Two numbers r and s sum up to \frac{359}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{359}{3} = \frac{359}{6}. 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>

$$r = \frac{359}{6} - u s = \frac{359}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 3540

$$(\frac{359}{6} - u) (\frac{359}{6} + u) = 3540$$

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

$$\frac{128881}{36} - u^2 = 3540$$

Simplify the expression by subtracting \frac{128881}{36} on both sides

$$-u^2 = 3540-\frac{128881}{36} = -\frac{1441}{36}$$

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

$$u^2 = \frac{1441}{36} u = \pm\sqrt{\frac{1441}{36}} = \pm \frac{\sqrt{1441}}{6}$$

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

$$r =\frac{359}{6} - \frac{\sqrt{1441}}{6} = 53.507 s = \frac{359}{6} + \frac{\sqrt{1441}}{6} = 66.160$$

Evaluate

$$3n^{2}-359n+10620$$