Solve 26x^2-572x+132

$$26 x ^ { 2 } - 572 x + 132$$

Factor

$$26\left(x-\left(-\frac{\sqrt{19591}}{13}+11\right)\right)\left(x-\left(\frac{\sqrt{19591}}{13}+11\right)\right)$$

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.

$$26x^{2}-572x+132=0$$

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.

$$x=\frac{-\left(-572\right)±\sqrt{\left(-572\right)^{2}-4\times 26\times 132}}{2\times 26}$$

Square -572.

$$x=\frac{-\left(-572\right)±\sqrt{327184-4\times 26\times 132}}{2\times 26}$$

Multiply -4 times 26.

$$x=\frac{-\left(-572\right)±\sqrt{327184-104\times 132}}{2\times 26}$$

Multiply -104 times 132.

$$x=\frac{-\left(-572\right)±\sqrt{327184-13728}}{2\times 26}$$

Add 327184 to -13728.

$$x=\frac{-\left(-572\right)±\sqrt{313456}}{2\times 26}$$

Take the square root of 313456.

$$x=\frac{-\left(-572\right)±4\sqrt{19591}}{2\times 26}$$

The opposite of -572 is 572.

$$x=\frac{572±4\sqrt{19591}}{2\times 26}$$

Multiply 2 times 26.

$$x=\frac{572±4\sqrt{19591}}{52}$$

Now solve the equation x=\frac{572±4\sqrt{19591}}{52} when ± is plus. Add 572 to 4\sqrt{19591}.

$$x=\frac{4\sqrt{19591}+572}{52}$$

Divide 572+4\sqrt{19591} by 52.

$$x=\frac{\sqrt{19591}}{13}+11$$

Now solve the equation x=\frac{572±4\sqrt{19591}}{52} when ± is minus. Subtract 4\sqrt{19591} from 572.

$$x=\frac{572-4\sqrt{19591}}{52}$$

Divide 572-4\sqrt{19591} by 52.

$$x=-\frac{\sqrt{19591}}{13}+11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11+\frac{\sqrt{19591}}{13} for x_{1} and 11-\frac{\sqrt{19591}}{13} for x_{2}.

$$26x^{2}-572x+132=26\left(x-\left(\frac{\sqrt{19591}}{13}+11\right)\right)\left(x-\left(-\frac{\sqrt{19591}}{13}+11\right)\right)$$

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 26

$$x ^ 2 -22x +\frac{66}{13} = 0$$

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

$$r + s = 22 rs = \frac{66}{13}$$

Two numbers r and s sum up to 22 exactly when the average of the two numbers is \frac{1}{2}*22 = 11. 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>

$$r = 11 - u s = 11 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{66}{13}

$$(11 - u) (11 + u) = \frac{66}{13}$$

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

$$121 - u^2 = \frac{66}{13}$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = \frac{66}{13}-121 = -\frac{1507}{13}$$

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

$$u^2 = \frac{1507}{13} u = \pm\sqrt{\frac{1507}{13}} = \pm \frac{\sqrt{1507}}{\sqrt{13}}$$

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

$$r =11 - \frac{\sqrt{1507}}{\sqrt{13}} = 0.233 s = 11 + \frac{\sqrt{1507}}{\sqrt{13}} = 21.767$$

Evaluate

$$26x^{2}-572x+132$$