Solve 27x^2-6x-1=0

$$27 { x }^{ 2 } -6x-1 = 0$$

Solve for x

$$x=-\frac{1}{9}\approx -0.111111111$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 27x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-6 ab=27\left(-1\right)=-27$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -27.

$$1,-27 3,-9$$

Calculate the sum for each pair.

$$1-27=-26 3-9=-6$$

The solution is the pair that gives sum -6.

$$a=-9 b=3$$

Rewrite 27x^{2}-6x-1 as \left(27x^{2}-9x\right)+\left(3x-1\right).

$$\left(27x^{2}-9x\right)+\left(3x-1\right)$$

Factor out 9x in 27x^{2}-9x.

$$9x\left(3x-1\right)+3x-1$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(9x+1\right)$$

To find equation solutions, solve 3x-1=0 and 9x+1=0.

$$x=\frac{1}{3} x=-\frac{1}{9}$$

Steps Using the Quadratic Formula

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.

$$27x^{2}-6x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 27\left(-1\right)}}{2\times 27}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 27\left(-1\right)}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-\left(-6\right)±\sqrt{36-108\left(-1\right)}}{2\times 27}$$

Multiply -108 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+108}}{2\times 27}$$

Add 36 to 108.

$$x=\frac{-\left(-6\right)±\sqrt{144}}{2\times 27}$$

Take the square root of 144.

$$x=\frac{-\left(-6\right)±12}{2\times 27}$$

The opposite of -6 is 6.

$$x=\frac{6±12}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{6±12}{54}$$

Now solve the equation x=\frac{6±12}{54} when ± is plus. Add 6 to 12.

$$x=\frac{18}{54}$$

Reduce the fraction \frac{18}{54} to lowest terms by extracting and canceling out 18.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{6±12}{54} when ± is minus. Subtract 12 from 6.

$$x=-\frac{6}{54}$$

Reduce the fraction \frac{-6}{54} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1}{9}$$

The equation is now solved.

$$x=\frac{1}{3} x=-\frac{1}{9}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$27x^{2}-6x-1=0$$

Add 1 to both sides of the equation.

$$27x^{2}-6x-1-\left(-1\right)=-\left(-1\right)$$

Subtracting -1 from itself leaves 0.

$$27x^{2}-6x=-\left(-1\right)$$

Subtract -1 from 0.

$$27x^{2}-6x=1$$

Divide both sides by 27.

$$\frac{27x^{2}-6x}{27}=\frac{1}{27}$$

Dividing by 27 undoes the multiplication by 27.

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

Reduce the fraction \frac{-6}{27} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{2}{9}x=\frac{1}{27}$$

Divide -\frac{2}{9}, the coefficient of the x term, by 2 to get -\frac{1}{9}. Then add the square of -\frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{9}x+\frac{1}{81}=\frac{1}{27}+\frac{1}{81}$$

Add \frac{1}{27} to \frac{1}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{9}x+\frac{1}{81}=\frac{4}{81}$$

Factor x^{2}-\frac{2}{9}x+\frac{1}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{9}\right)^{2}=\frac{4}{81}$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{1}{9}=\frac{2}{9} x-\frac{1}{9}=-\frac{2}{9}$$

Add \frac{1}{9} to both sides of the equation.

$$x=\frac{1}{3} x=-\frac{1}{9}$$