Solve 3n^2-357n+10620

$$3 n ^ { 2 } - 357 n + 10620$$

Factor

$$3\left(n-60\right)\left(n-59\right)$$

Solution Steps

Factor out 3.

$$3\left(n^{2}-119n+3540\right)$$

Consider n^{2}-119n+3540. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+3540. To find a and b, set up a system to be solved.

$$a+b=-119 ab=1\times 3540=3540$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 3540.

$$-1,-3540 -2,-1770 -3,-1180 -4,-885 -5,-708 -6,-590 -10,-354 -12,-295 -15,-236 -20,-177 -30,-118 -59,-60$$

Calculate the sum for each pair.

$$-1-3540=-3541 -2-1770=-1772 -3-1180=-1183 -4-885=-889 -5-708=-713 -6-590=-596 -10-354=-364 -12-295=-307 -15-236=-251 -20-177=-197 -30-118=-148 -59-60=-119$$

The solution is the pair that gives sum -119.

$$a=-60 b=-59$$

Rewrite n^{2}-119n+3540 as \left(n^{2}-60n\right)+\left(-59n+3540\right).

$$\left(n^{2}-60n\right)+\left(-59n+3540\right)$$

Factor out n in the first and -59 in the second group.

$$n\left(n-60\right)-59\left(n-60\right)$$

Factor out common term n-60 by using distributive property.

$$\left(n-60\right)\left(n-59\right)$$

Rewrite the complete factored expression.

$$3\left(n-60\right)\left(n-59\right)$$

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.

$$3n^{2}-357n+10620=0$$

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.

$$n=\frac{-\left(-357\right)±\sqrt{\left(-357\right)^{2}-4\times 3\times 10620}}{2\times 3}$$

Square -357.

$$n=\frac{-\left(-357\right)±\sqrt{127449-4\times 3\times 10620}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-357\right)±\sqrt{127449-12\times 10620}}{2\times 3}$$

Multiply -12 times 10620.

$$n=\frac{-\left(-357\right)±\sqrt{127449-127440}}{2\times 3}$$

Add 127449 to -127440.

$$n=\frac{-\left(-357\right)±\sqrt{9}}{2\times 3}$$

Take the square root of 9.

$$n=\frac{-\left(-357\right)±3}{2\times 3}$$

The opposite of -357 is 357.

$$n=\frac{357±3}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{357±3}{6}$$

Now solve the equation n=\frac{357±3}{6} when ± is plus. Add 357 to 3.

$$n=\frac{360}{6}$$

Divide 360 by 6.

$$n=60$$

Now solve the equation n=\frac{357±3}{6} when ± is minus. Subtract 3 from 357.

$$n=\frac{354}{6}$$

Divide 354 by 6.

$$n=59$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 60 for x_{1} and 59 for x_{2}.

$$3n^{2}-357n+10620=3\left(n-60\right)\left(n-59\right)$$

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

$$x ^ 2 -119x +3540 = 0$$

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

$$r + s = 119 rs = 3540$$

Two numbers r and s sum up to 119 exactly when the average of the two numbers is \frac{1}{2}*119 = \frac{119}{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>

$$r = \frac{119}{2} - u s = \frac{119}{2} + u$$

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

$$(\frac{119}{2} - u) (\frac{119}{2} + u) = 3540$$

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

$$\frac{14161}{4} - u^2 = 3540$$

Simplify the expression by subtracting \frac{14161}{4} on both sides

$$-u^2 = 3540-\frac{14161}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =\frac{119}{2} - \frac{1}{2} = 59 s = \frac{119}{2} + \frac{1}{2} = 60$$

Evaluate

$$3\left(n-60\right)\left(n-59\right)$$