Solve 36x^2-218x+210

$$36 x ^ { 2 } - 218 x + 210$$

Factor

$$36\left(x-\frac{109-\sqrt{4321}}{36}\right)\left(x-\frac{\sqrt{4321}+109}{36}\right)$$

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.

$$36x^{2}-218x+210=0$$

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.

$$x=\frac{-\left(-218\right)±\sqrt{\left(-218\right)^{2}-4\times 36\times 210}}{2\times 36}$$

Square -218.

$$x=\frac{-\left(-218\right)±\sqrt{47524-4\times 36\times 210}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-218\right)±\sqrt{47524-144\times 210}}{2\times 36}$$

Multiply -144 times 210.

$$x=\frac{-\left(-218\right)±\sqrt{47524-30240}}{2\times 36}$$

Add 47524 to -30240.

$$x=\frac{-\left(-218\right)±\sqrt{17284}}{2\times 36}$$

Take the square root of 17284.

$$x=\frac{-\left(-218\right)±2\sqrt{4321}}{2\times 36}$$

The opposite of -218 is 218.

$$x=\frac{218±2\sqrt{4321}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{218±2\sqrt{4321}}{72}$$

Now solve the equation x=\frac{218±2\sqrt{4321}}{72} when ± is plus. Add 218 to 2\sqrt{4321}.

$$x=\frac{2\sqrt{4321}+218}{72}$$

Divide 218+2\sqrt{4321} by 72.

$$x=\frac{\sqrt{4321}+109}{36}$$

Now solve the equation x=\frac{218±2\sqrt{4321}}{72} when ± is minus. Subtract 2\sqrt{4321} from 218.

$$x=\frac{218-2\sqrt{4321}}{72}$$

Divide 218-2\sqrt{4321} by 72.

$$x=\frac{109-\sqrt{4321}}{36}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{109+\sqrt{4321}}{36} for x_{1} and \frac{109-\sqrt{4321}}{36} for x_{2}.

$$36x^{2}-218x+210=36\left(x-\frac{\sqrt{4321}+109}{36}\right)\left(x-\frac{109-\sqrt{4321}}{36}\right)$$

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 36

$$x ^ 2 -\frac{109}{18}x +\frac{35}{6} = 0$$

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

$$r + s = \frac{109}{18} rs = \frac{35}{6}$$

Two numbers r and s sum up to \frac{109}{18} exactly when the average of the two numbers is \frac{1}{2}*\frac{109}{18} = \frac{109}{36}. 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>

$$r = \frac{109}{36} - u s = \frac{109}{36} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{35}{6}

$$(\frac{109}{36} - u) (\frac{109}{36} + u) = \frac{35}{6}$$

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

$$\frac{11881}{1296} - u^2 = \frac{35}{6}$$

Simplify the expression by subtracting \frac{11881}{1296} on both sides

$$-u^2 = \frac{35}{6}-\frac{11881}{1296} = -\frac{4321}{1296}$$

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

$$u^2 = \frac{4321}{1296} u = \pm\sqrt{\frac{4321}{1296}} = \pm \frac{\sqrt{4321}}{36}$$

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

$$r =\frac{109}{36} - \frac{\sqrt{4321}}{36} = 1.202 s = \frac{109}{36} + \frac{\sqrt{4321}}{36} = 4.854$$

Evaluate

$$36x^{2}-218x+210$$