Solve =7n^2+2407n-61800

$$= 7 n ^ { 2 } + 2407 n - 61800$$

Factor

$$\left(n-24\right)\left(7n+2575\right)$$

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.

$$7n^{2}+2407n-61800=0$$

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.

$$n=\frac{-2407±\sqrt{2407^{2}-4\times 7\left(-61800\right)}}{2\times 7}$$

Square 2407.

$$n=\frac{-2407±\sqrt{5793649-4\times 7\left(-61800\right)}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-2407±\sqrt{5793649-28\left(-61800\right)}}{2\times 7}$$

Multiply -28 times -61800.

$$n=\frac{-2407±\sqrt{5793649+1730400}}{2\times 7}$$

Add 5793649 to 1730400.

$$n=\frac{-2407±\sqrt{7524049}}{2\times 7}$$

Take the square root of 7524049.

$$n=\frac{-2407±2743}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{-2407±2743}{14}$$

Now solve the equation n=\frac{-2407±2743}{14} when ± is plus. Add -2407 to 2743.

$$n=\frac{336}{14}$$

Divide 336 by 14.

$$n=24$$

Now solve the equation n=\frac{-2407±2743}{14} when ± is minus. Subtract 2743 from -2407.

$$n=-\frac{5150}{14}$$

Reduce the fraction \frac{-5150}{14} to lowest terms by extracting and canceling out 2.

$$n=-\frac{2575}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24 for x_{1} and -\frac{2575}{7} for x_{2}.

$$7n^{2}+2407n-61800=7\left(n-24\right)\left(n-\left(-\frac{2575}{7}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$7n^{2}+2407n-61800=7\left(n-24\right)\left(n+\frac{2575}{7}\right)$$

Add \frac{2575}{7} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7n^{2}+2407n-61800=7\left(n-24\right)\times \frac{7n+2575}{7}$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7n^{2}+2407n-61800=\left(n-24\right)\left(7n+2575\right)$$

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 7

$$x ^ 2 +\frac{2407}{7}x -\frac{61800}{7} = 0$$

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

$$r + s = -\frac{2407}{7} rs = -\frac{61800}{7}$$

Two numbers r and s sum up to -\frac{2407}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2407}{7} = -\frac{2407}{14}. 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>

$$r = -\frac{2407}{14} - u s = -\frac{2407}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{61800}{7}

$$(-\frac{2407}{14} - u) (-\frac{2407}{14} + u) = -\frac{61800}{7}$$

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

$$\frac{5793649}{196} - u^2 = -\frac{61800}{7}$$

Simplify the expression by subtracting \frac{5793649}{196} on both sides

$$-u^2 = -\frac{61800}{7}-\frac{5793649}{196} = -\frac{7524049}{196}$$

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

$$u^2 = \frac{7524049}{196} u = \pm\sqrt{\frac{7524049}{196}} = \pm \frac{2743}{14}$$

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

$$r =-\frac{2407}{14} - \frac{2743}{14} = -367.857 s = -\frac{2407}{14} + \frac{2743}{14} = 24$$

Evaluate

$$\left(n-24\right)\left(7n+2575\right)$$