Solve C(x)=13x^2-66x+36

$$C ( x ) = 13 x ^ { 2 } - 66 x + 36$$

Factor

$$13\left(x-\frac{33-3\sqrt{69}}{13}\right)\left(x-\frac{3\sqrt{69}+33}{13}\right)$$

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.

$$13x^{2}-66x+36=0$$

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.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 13\times 36}}{2\times 13}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\times 13\times 36}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-66\right)±\sqrt{4356-52\times 36}}{2\times 13}$$

Multiply -52 times 36.

$$x=\frac{-\left(-66\right)±\sqrt{4356-1872}}{2\times 13}$$

Add 4356 to -1872.

$$x=\frac{-\left(-66\right)±\sqrt{2484}}{2\times 13}$$

Take the square root of 2484.

$$x=\frac{-\left(-66\right)±6\sqrt{69}}{2\times 13}$$

The opposite of -66 is 66.

$$x=\frac{66±6\sqrt{69}}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{66±6\sqrt{69}}{26}$$

Now solve the equation x=\frac{66±6\sqrt{69}}{26} when ± is plus. Add 66 to 6\sqrt{69}.

$$x=\frac{6\sqrt{69}+66}{26}$$

Divide 66+6\sqrt{69} by 26.

$$x=\frac{3\sqrt{69}+33}{13}$$

Now solve the equation x=\frac{66±6\sqrt{69}}{26} when ± is minus. Subtract 6\sqrt{69} from 66.

$$x=\frac{66-6\sqrt{69}}{26}$$

Divide 66-6\sqrt{69} by 26.

$$x=\frac{33-3\sqrt{69}}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{33+3\sqrt{69}}{13} for x_{1} and \frac{33-3\sqrt{69}}{13} for x_{2}.

$$13x^{2}-66x+36=13\left(x-\frac{3\sqrt{69}+33}{13}\right)\left(x-\frac{33-3\sqrt{69}}{13}\right)$$

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 13

$$x ^ 2 -\frac{66}{13}x +\frac{36}{13} = 0$$

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

$$r + s = \frac{66}{13} rs = \frac{36}{13}$$

Two numbers r and s sum up to \frac{66}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{66}{13} = \frac{33}{13}. 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>

$$r = \frac{33}{13} - u s = \frac{33}{13} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{36}{13}

$$(\frac{33}{13} - u) (\frac{33}{13} + u) = \frac{36}{13}$$

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

$$\frac{1089}{169} - u^2 = \frac{36}{13}$$

Simplify the expression by subtracting \frac{1089}{169} on both sides

$$-u^2 = \frac{36}{13}-\frac{1089}{169} = -\frac{621}{169}$$

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

$$u^2 = \frac{621}{169} u = \pm\sqrt{\frac{621}{169}} = \pm \frac{\sqrt{621}}{13}$$

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

$$r =\frac{33}{13} - \frac{\sqrt{621}}{13} = 0.622 s = \frac{33}{13} + \frac{\sqrt{621}}{13} = 4.455$$

Evaluate

$$13x^{2}-66x+36$$