Solve by Completing the Square 49x^2-84x=-29

$49 x^{2} - 84 x = - 29$

Step 1

Divide each term in $49 x^{2} - 84 x = - 29$ by $49$ and simplify.

Divide each term in $49 x^{2} - 84 x = - 29$ by $49$ .

$\frac{49 x^{2}}{49} + \frac{- 84 x}{49} = \frac{- 29}{49}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 x^{2}}{49} + \frac{- 84 x}{49} = \frac{- 29}{49}$

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

$x^{2} + \frac{- 84 x}{49} = \frac{- 29}{49}$

Cancel the common factor of $- 84$ and $49$ .

Factor $7$ out of $- 84 x$ .

$x^{2} + \frac{7 (- 12 x)}{49} = \frac{- 29}{49}$

Cancel the common factors.

Factor $7$ out of $49$ .

$x^{2} + \frac{7 (- 12 x)}{7 (7)} = \frac{- 29}{49}$

Cancel the common factor.

$x^{2} + \frac{7 (- 12 x)}{7 \cdot 7} = \frac{- 29}{49}$

Rewrite the expression.

$x^{2} + \frac{- 12 x}{7} = \frac{- 29}{49}$

Move the negative in front of the fraction.

$x^{2} - \frac{12 x}{7} = \frac{- 29}{49}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} - \frac{12 x}{7} = - \frac{29}{49}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{6}{7})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{12 x}{7} + {(- \frac{6}{7})}^{2} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

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 $- \frac{6}{7}$ .

$x^{2} - \frac{12 x}{7} + {(- 1)}^{2} {(\frac{6}{7})}^{2} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

Apply the product rule to $\frac{6}{7}$ .

$x^{2} - \frac{12 x}{7} + {(- 1)}^{2} \frac{6^{2}}{7^{2}} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

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

$x^{2} - \frac{12 x}{7} + 1 \frac{6^{2}}{7^{2}} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

Multiply $\frac{6^{2}}{7^{2}}$ by $1$ .

$x^{2} - \frac{12 x}{7} + \frac{6^{2}}{7^{2}} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

Raise $6$ to the power of $2$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{7^{2}} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + {(- \frac{6}{7})}^{2}$

Simplify the right side.

Simplify $- \frac{29}{49} + {(- \frac{6}{7})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{6}{7}$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + {(- 1)}^{2} {(\frac{6}{7})}^{2}$

Apply the product rule to $\frac{6}{7}$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + {(- 1)}^{2} \frac{6^{2}}{7^{2}}$

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

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + 1 \frac{6^{2}}{7^{2}}$

Multiply $\frac{6^{2}}{7^{2}}$ by $1$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + \frac{6^{2}}{7^{2}}$

Raise $6$ to the power of $2$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + \frac{36}{7^{2}}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = - \frac{29}{49} + \frac{36}{49}$

Simplify terms.

Combine the numerators over the common denominator.

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{- 29 + 36}{49}$

Add $- 29$ and $36$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{7}{49}$

Cancel the common factor of $7$ and $49$ .

Factor $7$ out of $7$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{7 (1)}{49}$

Cancel the common factors.

Factor $7$ out of $49$ .

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{7 \cdot 1}{7 \cdot 7}$

Cancel the common factor.

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{7 \cdot 1}{7 \cdot 7}$

Rewrite the expression.

$x^{2} - \frac{12 x}{7} + \frac{36}{49} = \frac{1}{7}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{6}{7})}^{2}$ .

${(x - \frac{6}{7})}^{2} = \frac{1}{7}$

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.

$x - \frac{6}{7} = \pm \sqrt{\frac{1}{7}}$

Simplify $\pm \sqrt{\frac{1}{7}}$ .

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{1}}{\sqrt{7}}$

Any root of $1$ is $1$ .

$x - \frac{6}{7} = \pm \frac{1}{\sqrt{7}}$

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x - \frac{6}{7} = \pm \frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{7^{1}}$

Evaluate the exponent.

$x - \frac{6}{7} = \pm \frac{\sqrt{7}}{7}$

Add $\frac{6}{7}$ to both sides of the equation.

$x = \pm \frac{\sqrt{7}}{7} + \frac{6}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{7}}{7} + \frac{6}{7}$

Decimal Form:

$x = 1.23510733 \dots, 0.47917838 \dots$

Solve by Completing the Square 49x^2-84x=-29

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