Solve 25n^2+2025n-83100

$$25 n ^ { 2 } + 2025 n - 83100$$

Factor

$$25\left(n-\frac{-\sqrt{19857}-81}{2}\right)\left(n-\frac{\sqrt{19857}-81}{2}\right)$$

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.

$$25n^{2}+2025n-83100=0$$

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.

$$n=\frac{-2025±\sqrt{2025^{2}-4\times 25\left(-83100\right)}}{2\times 25}$$

Square 2025.

$$n=\frac{-2025±\sqrt{4100625-4\times 25\left(-83100\right)}}{2\times 25}$$

Multiply -4 times 25.

$$n=\frac{-2025±\sqrt{4100625-100\left(-83100\right)}}{2\times 25}$$

Multiply -100 times -83100.

$$n=\frac{-2025±\sqrt{4100625+8310000}}{2\times 25}$$

Add 4100625 to 8310000.

$$n=\frac{-2025±\sqrt{12410625}}{2\times 25}$$

Take the square root of 12410625.

$$n=\frac{-2025±25\sqrt{19857}}{2\times 25}$$

Multiply 2 times 25.

$$n=\frac{-2025±25\sqrt{19857}}{50}$$

Now solve the equation n=\frac{-2025±25\sqrt{19857}}{50} when ± is plus. Add -2025 to 25\sqrt{19857}.

$$n=\frac{25\sqrt{19857}-2025}{50}$$

Divide -2025+25\sqrt{19857} by 50.

$$n=\frac{\sqrt{19857}-81}{2}$$

Now solve the equation n=\frac{-2025±25\sqrt{19857}}{50} when ± is minus. Subtract 25\sqrt{19857} from -2025.

$$n=\frac{-25\sqrt{19857}-2025}{50}$$

Divide -2025-25\sqrt{19857} by 50.

$$n=\frac{-\sqrt{19857}-81}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-81+\sqrt{19857}}{2} for x_{1} and \frac{-81-\sqrt{19857}}{2} for x_{2}.

$$25n^{2}+2025n-83100=25\left(n-\frac{\sqrt{19857}-81}{2}\right)\left(n-\frac{-\sqrt{19857}-81}{2}\right)$$

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 25

$$x ^ 2 +81x -3324 = 0$$

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

$$r + s = -81 rs = -3324$$

Two numbers r and s sum up to -81 exactly when the average of the two numbers is \frac{1}{2}*-81 = -\frac{81}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{81}{2} - u s = -\frac{81}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -3324

$$(-\frac{81}{2} - u) (-\frac{81}{2} + u) = -3324$$

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

$$\frac{6561}{4} - u^2 = -3324$$

Simplify the expression by subtracting \frac{6561}{4} on both sides

$$-u^2 = -3324-\frac{6561}{4} = -\frac{19857}{4}$$

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

$$u^2 = \frac{19857}{4} u = \pm\sqrt{\frac{19857}{4}} = \pm \frac{\sqrt{19857}}{2}$$

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

$$r =-\frac{81}{2} - \frac{\sqrt{19857}}{2} = -110.957 s = -\frac{81}{2} + \frac{\sqrt{19857}}{2} = 29.957$$

Evaluate

$$25\left(n^{2}+81n-3324\right)$$