Solve 25a^2-520a-2860

$$25 a ^ { 2 } - 520 a - 2860$$

Factor

$$25\left(a-\frac{52-2\sqrt{1391}}{5}\right)\left(a-\frac{2\sqrt{1391}+52}{5}\right)$$

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.

$$25a^{2}-520a-2860=0$$

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.

$$a=\frac{-\left(-520\right)±\sqrt{\left(-520\right)^{2}-4\times 25\left(-2860\right)}}{2\times 25}$$

Square -520.

$$a=\frac{-\left(-520\right)±\sqrt{270400-4\times 25\left(-2860\right)}}{2\times 25}$$

Multiply -4 times 25.

$$a=\frac{-\left(-520\right)±\sqrt{270400-100\left(-2860\right)}}{2\times 25}$$

Multiply -100 times -2860.

$$a=\frac{-\left(-520\right)±\sqrt{270400+286000}}{2\times 25}$$

Add 270400 to 286000.

$$a=\frac{-\left(-520\right)±\sqrt{556400}}{2\times 25}$$

Take the square root of 556400.

$$a=\frac{-\left(-520\right)±20\sqrt{1391}}{2\times 25}$$

The opposite of -520 is 520.

$$a=\frac{520±20\sqrt{1391}}{2\times 25}$$

Multiply 2 times 25.

$$a=\frac{520±20\sqrt{1391}}{50}$$

Now solve the equation a=\frac{520±20\sqrt{1391}}{50} when ± is plus. Add 520 to 20\sqrt{1391}.

$$a=\frac{20\sqrt{1391}+520}{50}$$

Divide 520+20\sqrt{1391} by 50.

$$a=\frac{2\sqrt{1391}+52}{5}$$

Now solve the equation a=\frac{520±20\sqrt{1391}}{50} when ± is minus. Subtract 20\sqrt{1391} from 520.

$$a=\frac{520-20\sqrt{1391}}{50}$$

Divide 520-20\sqrt{1391} by 50.

$$a=\frac{52-2\sqrt{1391}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{52+2\sqrt{1391}}{5} for x_{1} and \frac{52-2\sqrt{1391}}{5} for x_{2}.

$$25a^{2}-520a-2860=25\left(a-\frac{2\sqrt{1391}+52}{5}\right)\left(a-\frac{52-2\sqrt{1391}}{5}\right)$$

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

$$x ^ 2 -\frac{104}{5}x -\frac{572}{5} = 0$$

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

$$r + s = \frac{104}{5} rs = -\frac{572}{5}$$

Two numbers r and s sum up to \frac{104}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{104}{5} = \frac{52}{5}. 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>

$$r = \frac{52}{5} - u s = \frac{52}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{572}{5}

$$(\frac{52}{5} - u) (\frac{52}{5} + u) = -\frac{572}{5}$$

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

$$\frac{2704}{25} - u^2 = -\frac{572}{5}$$

Simplify the expression by subtracting \frac{2704}{25} on both sides

$$-u^2 = -\frac{572}{5}-\frac{2704}{25} = -\frac{5564}{25}$$

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

$$u^2 = \frac{5564}{25} u = \pm\sqrt{\frac{5564}{25}} = \pm \frac{\sqrt{5564}}{5}$$

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

$$r =\frac{52}{5} - \frac{\sqrt{5564}}{5} = -4.518 s = \frac{52}{5} + \frac{\sqrt{5564}}{5} = 25.318$$

Evaluate

$$25a^{2}-520a-2860$$