Solve 31a^2-837a-7290

$$31 a ^ { 2 } - 837 a - 7290$$

Factor

$$31\left(a-\left(-\frac{27\sqrt{2201}}{62}+\frac{27}{2}\right)\right)\left(a-\left(\frac{27\sqrt{2201}}{62}+\frac{27}{2}\right)\right)$$

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.

$$31a^{2}-837a-7290=0$$

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.

$$a=\frac{-\left(-837\right)±\sqrt{\left(-837\right)^{2}-4\times 31\left(-7290\right)}}{2\times 31}$$

Square -837.

$$a=\frac{-\left(-837\right)±\sqrt{700569-4\times 31\left(-7290\right)}}{2\times 31}$$

Multiply -4 times 31.

$$a=\frac{-\left(-837\right)±\sqrt{700569-124\left(-7290\right)}}{2\times 31}$$

Multiply -124 times -7290.

$$a=\frac{-\left(-837\right)±\sqrt{700569+903960}}{2\times 31}$$

Add 700569 to 903960.

$$a=\frac{-\left(-837\right)±\sqrt{1604529}}{2\times 31}$$

Take the square root of 1604529.

$$a=\frac{-\left(-837\right)±27\sqrt{2201}}{2\times 31}$$

The opposite of -837 is 837.

$$a=\frac{837±27\sqrt{2201}}{2\times 31}$$

Multiply 2 times 31.

$$a=\frac{837±27\sqrt{2201}}{62}$$

Now solve the equation a=\frac{837±27\sqrt{2201}}{62} when ± is plus. Add 837 to 27\sqrt{2201}.

$$a=\frac{27\sqrt{2201}+837}{62}$$

Divide 837+27\sqrt{2201} by 62.

$$a=\frac{27\sqrt{2201}}{62}+\frac{27}{2}$$

Now solve the equation a=\frac{837±27\sqrt{2201}}{62} when ± is minus. Subtract 27\sqrt{2201} from 837.

$$a=\frac{837-27\sqrt{2201}}{62}$$

Divide 837-27\sqrt{2201} by 62.

$$a=-\frac{27\sqrt{2201}}{62}+\frac{27}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{27}{2}+\frac{27\sqrt{2201}}{62} for x_{1} and \frac{27}{2}-\frac{27\sqrt{2201}}{62} for x_{2}.

$$31a^{2}-837a-7290=31\left(a-\left(\frac{27\sqrt{2201}}{62}+\frac{27}{2}\right)\right)\left(a-\left(-\frac{27\sqrt{2201}}{62}+\frac{27}{2}\right)\right)$$

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 31

$$x ^ 2 -27x -\frac{7290}{31} = 0$$

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

$$r + s = 27 rs = -\frac{7290}{31}$$

Two numbers r and s sum up to 27 exactly when the average of the two numbers is \frac{1}{2}*27 = \frac{27}{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>

$$r = \frac{27}{2} - u s = \frac{27}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7290}{31}

$$(\frac{27}{2} - u) (\frac{27}{2} + u) = -\frac{7290}{31}$$

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

$$\frac{729}{4} - u^2 = -\frac{7290}{31}$$

Simplify the expression by subtracting \frac{729}{4} on both sides

$$-u^2 = -\frac{7290}{31}-\frac{729}{4} = -\frac{51759}{124}$$

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

$$u^2 = \frac{51759}{124} u = \pm\sqrt{\frac{51759}{124}} = \pm \frac{\sqrt{51759}}{\sqrt{124}}$$

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

$$r =\frac{27}{2} - \frac{\sqrt{51759}}{\sqrt{124}} = -6.931 s = \frac{27}{2} + \frac{\sqrt{51759}}{\sqrt{124}} = 33.931$$

Evaluate

$$31a^{2}-837a-7290$$