Solve by Completing the Square 6x^2-x-5=0

$6 x^{2} - x - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$6 x^{2} - x = 5$

Step 2

Divide each term in $6 x^{2} - x = 5$ by $6$ and simplify.

Divide each term in $6 x^{2} - x = 5$ by $6$ .

$\frac{6 x^{2}}{6} + \frac{- x}{6} = \frac{5}{6}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{2}}{6} + \frac{- x}{6} = \frac{5}{6}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{- x}{6} = \frac{5}{6}$

Move the negative in front of the fraction.

$x^{2} - \frac{x}{6} = \frac{5}{6}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{1}{12})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{x}{6} + {(- \frac{1}{12})}^{2} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{12}$ .

$x^{2} - \frac{x}{6} + {(- 1)}^{2} {(\frac{1}{12})}^{2} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Apply the product rule to $\frac{1}{12}$ .

$x^{2} - \frac{x}{6} + {(- 1)}^{2} \frac{1^{2}}{12^{2}} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{x}{6} + 1 \frac{1^{2}}{12^{2}} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Multiply $\frac{1^{2}}{12^{2}}$ by $1$ .

$x^{2} - \frac{x}{6} + \frac{1^{2}}{12^{2}} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

One to any power is one.

$x^{2} - \frac{x}{6} + \frac{1}{12^{2}} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Raise $12$ to the power of $2$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + {(- \frac{1}{12})}^{2}$

Simplify the right side.

Simplify $\frac{5}{6} + {(- \frac{1}{12})}^{2}$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{12}$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + {(- 1)}^{2} {(\frac{1}{12})}^{2}$

Apply the product rule to $\frac{1}{12}$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + {(- 1)}^{2} \frac{1^{2}}{12^{2}}$

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + 1 \frac{1^{2}}{12^{2}}$

Multiply $\frac{1^{2}}{12^{2}}$ by $1$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + \frac{1^{2}}{12^{2}}$

One to any power is one.

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + \frac{1}{12^{2}}$

Raise $12$ to the power of $2$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} + \frac{1}{144}$

To write $\frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5}{6} \cdot \frac{24}{24} + \frac{1}{144}$

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{6}$ by $\frac{24}{24}$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5 \cdot 24}{6 \cdot 24} + \frac{1}{144}$

Multiply $6$ by $24$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5 \cdot 24}{144} + \frac{1}{144}$

Combine the numerators over the common denominator.

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{5 \cdot 24 + 1}{144}$

Simplify the numerator.

Multiply $5$ by $24$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{120 + 1}{144}$

Add $120$ and $1$ .

$x^{2} - \frac{x}{6} + \frac{1}{144} = \frac{121}{144}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{1}{12})}^{2}$ .

${(x - \frac{1}{12})}^{2} = \frac{121}{144}$

Step 7

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - \frac{1}{12} = \pm \sqrt{\frac{121}{144}}$

Simplify $\pm \sqrt{\frac{121}{144}}$ .

Rewrite $\sqrt{\frac{121}{144}}$ as $\frac{\sqrt{121}}{\sqrt{144}}$ .

$x - \frac{1}{12} = \pm \frac{\sqrt{121}}{\sqrt{144}}$

Simplify the numerator.

Rewrite $121$ as $11^{2}$ .

$x - \frac{1}{12} = \pm \frac{\sqrt{11^{2}}}{\sqrt{144}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{1}{12} = \pm \frac{11}{\sqrt{144}}$

Simplify the denominator.

Rewrite $144$ as $12^{2}$ .

$x - \frac{1}{12} = \pm \frac{11}{\sqrt{12^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{1}{12} = \pm \frac{11}{12}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - \frac{1}{12} = \frac{11}{12}$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{12}$ to both sides of the equation.

$x = \frac{11}{12} + \frac{1}{12}$

Combine the numerators over the common denominator.

$x = \frac{11 + 1}{12}$

Add $11$ and $1$ .

$x = \frac{12}{12}$

Divide $12$ by $12$ .

$x = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$x - \frac{1}{12} = - \frac{11}{12}$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{12}$ to both sides of the equation.

$x = - \frac{11}{12} + \frac{1}{12}$

Combine the numerators over the common denominator.

$x = \frac{- 11 + 1}{12}$

Add $- 11$ and $1$ .

$x = \frac{- 10}{12}$

Cancel the common factor of $- 10$ and $12$ .

Factor $2$ out of $- 10$ .

$x = \frac{2 (- 5)}{12}$

Cancel the common factors.

Factor $2$ out of $12$ .

$x = \frac{2 \cdot - 5}{2 \cdot 6}$

Cancel the common factor.

$x = \frac{2 \cdot - 5}{2 \cdot 6}$

Rewrite the expression.

$x = \frac{- 5}{6}$

Move the negative in front of the fraction.

$x = - \frac{5}{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - \frac{5}{6}$

Solve by Completing the Square 6x^2-x-5=0

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