Solve 202x^2-108109x+2491

$$202 x ^ { 2 } - 108109 x + 2491$$

Factor

$$202\left(x-\frac{108109-\sqrt{11685543153}}{404}\right)\left(x-\frac{\sqrt{11685543153}+108109}{404}\right)$$

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.

$$202x^{2}-108109x+2491=0$$

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.

$$x=\frac{-\left(-108109\right)±\sqrt{\left(-108109\right)^{2}-4\times 202\times 2491}}{2\times 202}$$

Square -108109.

$$x=\frac{-\left(-108109\right)±\sqrt{11687555881-4\times 202\times 2491}}{2\times 202}$$

Multiply -4 times 202.

$$x=\frac{-\left(-108109\right)±\sqrt{11687555881-808\times 2491}}{2\times 202}$$

Multiply -808 times 2491.

$$x=\frac{-\left(-108109\right)±\sqrt{11687555881-2012728}}{2\times 202}$$

Add 11687555881 to -2012728.

$$x=\frac{-\left(-108109\right)±\sqrt{11685543153}}{2\times 202}$$

The opposite of -108109 is 108109.

$$x=\frac{108109±\sqrt{11685543153}}{2\times 202}$$

Multiply 2 times 202.

$$x=\frac{108109±\sqrt{11685543153}}{404}$$

Now solve the equation x=\frac{108109±\sqrt{11685543153}}{404} when ± is plus. Add 108109 to \sqrt{11685543153}.

$$x=\frac{\sqrt{11685543153}+108109}{404}$$

Now solve the equation x=\frac{108109±\sqrt{11685543153}}{404} when ± is minus. Subtract \sqrt{11685543153} from 108109.

$$x=\frac{108109-\sqrt{11685543153}}{404}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{108109+\sqrt{11685543153}}{404} for x_{1} and \frac{108109-\sqrt{11685543153}}{404} for x_{2}.

$$202x^{2}-108109x+2491=202\left(x-\frac{\sqrt{11685543153}+108109}{404}\right)\left(x-\frac{108109-\sqrt{11685543153}}{404}\right)$$

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 202

$$x ^ 2 -\frac{108109}{202}x +\frac{2491}{202} = 0$$

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

$$r + s = \frac{108109}{202} rs = \frac{2491}{202}$$

Two numbers r and s sum up to \frac{108109}{202} exactly when the average of the two numbers is \frac{1}{2}*\frac{108109}{202} = \frac{108109}{404}. 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>

$$r = \frac{108109}{404} - u s = \frac{108109}{404} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2491}{202}

$$(\frac{108109}{404} - u) (\frac{108109}{404} + u) = \frac{2491}{202}$$

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

$$-\frac{1197346007}{163216} - u^2 = \frac{2491}{202}$$

Simplify the expression by subtracting -\frac{1197346007}{163216} on both sides

$$-u^2 = \frac{2491}{202}--\frac{1197346007}{163216} = -\frac{1199358735}{163216}$$

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

$$u^2 = \frac{1199358735}{163216} u = \pm\sqrt{\frac{1199358735}{163216}} = \pm \frac{\sqrt{1199358735}}{404}$$

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

$$r =\frac{108109}{404} - \frac{\sqrt{1199358735}}{404} = 0.023 s = \frac{108109}{404} + \frac{\sqrt{1199358735}}{404} = 535.170$$

Evaluate

$$202x^{2}-108109x+2491$$