Solve 27x^2-9x-1

$$27 x ^ { 2 } - 9 x - 1$$

Factor

$$27\left(x-\left(-\frac{\sqrt{21}}{18}+\frac{1}{6}\right)\right)\left(x-\left(\frac{\sqrt{21}}{18}+\frac{1}{6}\right)\right)$$

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.

$$27x^{2}-9x-1=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 27\left(-1\right)}}{2\times 27}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 27\left(-1\right)}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-\left(-9\right)±\sqrt{81-108\left(-1\right)}}{2\times 27}$$

Multiply -108 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+108}}{2\times 27}$$

Add 81 to 108.

$$x=\frac{-\left(-9\right)±\sqrt{189}}{2\times 27}$$

Take the square root of 189.

$$x=\frac{-\left(-9\right)±3\sqrt{21}}{2\times 27}$$

The opposite of -9 is 9.

$$x=\frac{9±3\sqrt{21}}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{9±3\sqrt{21}}{54}$$

Now solve the equation x=\frac{9±3\sqrt{21}}{54} when ± is plus. Add 9 to 3\sqrt{21}.

$$x=\frac{3\sqrt{21}+9}{54}$$

Divide 9+3\sqrt{21} by 54.

$$x=\frac{\sqrt{21}}{18}+\frac{1}{6}$$

Now solve the equation x=\frac{9±3\sqrt{21}}{54} when ± is minus. Subtract 3\sqrt{21} from 9.

$$x=\frac{9-3\sqrt{21}}{54}$$

Divide 9-3\sqrt{21} by 54.

$$x=-\frac{\sqrt{21}}{18}+\frac{1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{6}+\frac{\sqrt{21}}{18} for x_{1} and \frac{1}{6}-\frac{\sqrt{21}}{18} for x_{2}.

$$27x^{2}-9x-1=27\left(x-\left(\frac{\sqrt{21}}{18}+\frac{1}{6}\right)\right)\left(x-\left(-\frac{\sqrt{21}}{18}+\frac{1}{6}\right)\right)$$

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 27

$$x ^ 2 -\frac{1}{3}x -\frac{1}{27} = 0$$

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

$$r + s = \frac{1}{3} rs = -\frac{1}{27}$$

Two numbers r and s sum up to \frac{1}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{3} = \frac{1}{6}. 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>

$$r = \frac{1}{6} - u s = \frac{1}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{27}

$$(\frac{1}{6} - u) (\frac{1}{6} + u) = -\frac{1}{27}$$

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

$$\frac{1}{36} - u^2 = -\frac{1}{27}$$

Simplify the expression by subtracting \frac{1}{36} on both sides

$$-u^2 = -\frac{1}{27}-\frac{1}{36} = -\frac{7}{108}$$

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

$$u^2 = \frac{7}{108} u = \pm\sqrt{\frac{7}{108}} = \pm \frac{\sqrt{7}}{\sqrt{108}}$$

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

$$r =\frac{1}{6} - \frac{\sqrt{7}}{\sqrt{108}} = -0.088 s = \frac{1}{6} + \frac{\sqrt{7}}{\sqrt{108}} = 0.421$$

Evaluate

$$27x^{2}-9x-1$$