Solve by Completing the Square 121x^2-110x=4

$121 x^{2} - 110 x = 4$

Step 1

Divide each term in $121 x^{2} - 110 x = 4$ by $121$ and simplify.

Divide each term in $121 x^{2} - 110 x = 4$ by $121$ .

$\frac{121 x^{2}}{121} + \frac{- 110 x}{121} = \frac{4}{121}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $121$ .

Cancel the common factor.

$\frac{121 x^{2}}{121} + \frac{- 110 x}{121} = \frac{4}{121}$

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

$x^{2} + \frac{- 110 x}{121} = \frac{4}{121}$

Cancel the common factor of $- 110$ and $121$ .

Factor $11$ out of $- 110 x$ .

$x^{2} + \frac{11 (- 10 x)}{121} = \frac{4}{121}$

Cancel the common factors.

Factor $11$ out of $121$ .

$x^{2} + \frac{11 (- 10 x)}{11 (11)} = \frac{4}{121}$

Cancel the common factor.

$x^{2} + \frac{11 (- 10 x)}{11 \cdot 11} = \frac{4}{121}$

Rewrite the expression.

$x^{2} + \frac{- 10 x}{11} = \frac{4}{121}$

Move the negative in front of the fraction.

$x^{2} - \frac{10 x}{11} = \frac{4}{121}$

Step 2

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

${(\frac{b}{2})}^{2} = {(- \frac{5}{11})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{10 x}{11} + {(- \frac{5}{11})}^{2} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Step 4

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 $- \frac{5}{11}$ .

$x^{2} - \frac{10 x}{11} + {(- 1)}^{2} {(\frac{5}{11})}^{2} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Apply the product rule to $\frac{5}{11}$ .

$x^{2} - \frac{10 x}{11} + {(- 1)}^{2} \frac{5^{2}}{11^{2}} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

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

$x^{2} - \frac{10 x}{11} + 1 \frac{5^{2}}{11^{2}} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Multiply $\frac{5^{2}}{11^{2}}$ by $1$ .

$x^{2} - \frac{10 x}{11} + \frac{5^{2}}{11^{2}} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Raise $5$ to the power of $2$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{11^{2}} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Raise $11$ to the power of $2$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + {(- \frac{5}{11})}^{2}$

Simplify the right side.

Simplify $\frac{4}{121} + {(- \frac{5}{11})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{5}{11}$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + {(- 1)}^{2} {(\frac{5}{11})}^{2}$

Apply the product rule to $\frac{5}{11}$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + {(- 1)}^{2} \frac{5^{2}}{11^{2}}$

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

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + 1 \frac{5^{2}}{11^{2}}$

Multiply $\frac{5^{2}}{11^{2}}$ by $1$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + \frac{5^{2}}{11^{2}}$

Raise $5$ to the power of $2$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + \frac{25}{11^{2}}$

Raise $11$ to the power of $2$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4}{121} + \frac{25}{121}$

Combine fractions.

Combine the numerators over the common denominator.

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{4 + 25}{121}$

Add $4$ and $25$ .

$x^{2} - \frac{10 x}{11} + \frac{25}{121} = \frac{29}{121}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{5}{11})}^{2}$ .

${(x - \frac{5}{11})}^{2} = \frac{29}{121}$

Step 6

Solve the equation for $x$ .

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

$x - \frac{5}{11} = \pm \sqrt{\frac{29}{121}}$

Simplify $\pm \sqrt{\frac{29}{121}}$ .

Rewrite $\sqrt{\frac{29}{121}}$ as $\frac{\sqrt{29}}{\sqrt{121}}$ .

$x - \frac{5}{11} = \pm \frac{\sqrt{29}}{\sqrt{121}}$

Simplify the denominator.

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

$x - \frac{5}{11} = \pm \frac{\sqrt{29}}{\sqrt{11^{2}}}$

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

$x - \frac{5}{11} = \pm \frac{\sqrt{29}}{11}$

Add $\frac{5}{11}$ to both sides of the equation.

$x = \pm \frac{\sqrt{29}}{11} + \frac{5}{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{29}}{11} + \frac{5}{11}$

Decimal Form:

$x = 0.94410589 \dots, - 0.03501498 \dots$

Solve by Completing the Square 121x^2-110x=4

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