Solve n^2+2401n-67400

$$n ^ { 2 } + 2401 n - 67400$$

Factor

$$\left(n-\frac{-3\sqrt{670489}-2401}{2}\right)\left(n-\frac{3\sqrt{670489}-2401}{2}\right)$$

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.

$$n^{2}+2401n-67400=0$$

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.

$$n=\frac{-2401±\sqrt{2401^{2}-4\left(-67400\right)}}{2}$$

Square 2401.

$$n=\frac{-2401±\sqrt{5764801-4\left(-67400\right)}}{2}$$

Multiply -4 times -67400.

$$n=\frac{-2401±\sqrt{5764801+269600}}{2}$$

Add 5764801 to 269600.

$$n=\frac{-2401±\sqrt{6034401}}{2}$$

Take the square root of 6034401.

$$n=\frac{-2401±3\sqrt{670489}}{2}$$

Now solve the equation n=\frac{-2401±3\sqrt{670489}}{2} when ± is plus. Add -2401 to 3\sqrt{670489}.

$$n=\frac{3\sqrt{670489}-2401}{2}$$

Now solve the equation n=\frac{-2401±3\sqrt{670489}}{2} when ± is minus. Subtract 3\sqrt{670489} from -2401.

$$n=\frac{-3\sqrt{670489}-2401}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2401+3\sqrt{670489}}{2} for x_{1} and \frac{-2401-3\sqrt{670489}}{2} for x_{2}.

$$n^{2}+2401n-67400=\left(n-\frac{3\sqrt{670489}-2401}{2}\right)\left(n-\frac{-3\sqrt{670489}-2401}{2}\right)$$

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.

$$x ^ 2 +2401x -67400 = 0$$

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

$$r + s = -2401 rs = -67400$$

Two numbers r and s sum up to -2401 exactly when the average of the two numbers is \frac{1}{2}*-2401 = -\frac{2401}{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>

$$r = -\frac{2401}{2} - u s = -\frac{2401}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -67400

$$(-\frac{2401}{2} - u) (-\frac{2401}{2} + u) = -67400$$

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

$$\frac{5764801}{4} - u^2 = -67400$$

Simplify the expression by subtracting \frac{5764801}{4} on both sides

$$-u^2 = -67400-\frac{5764801}{4} = -\frac{6034401}{4}$$

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

$$u^2 = \frac{6034401}{4} u = \pm\sqrt{\frac{6034401}{4}} = \pm \frac{\sqrt{6034401}}{2}$$

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

$$r =-\frac{2401}{2} - \frac{\sqrt{6034401}}{2} = -2428.751 s = -\frac{2401}{2} + \frac{\sqrt{6034401}}{2} = 27.751$$

Evaluate

$$n^{2}+2401n-67400$$