Solve by Completing the Square 7x^2-9x=14

$7 x^{2} - 9 x = 14$

Step 1

Divide each term in $7 x^{2} - 9 x = 14$ by $7$ and simplify.

Divide each term in $7 x^{2} - 9 x = 14$ by $7$ .

$\frac{7 x^{2}}{7} + \frac{- 9 x}{7} = \frac{14}{7}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{2}}{7} + \frac{- 9 x}{7} = \frac{14}{7}$

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

$x^{2} + \frac{- 9 x}{7} = \frac{14}{7}$

Move the negative in front of the fraction.

$x^{2} - \frac{9 x}{7} = \frac{14}{7}$

Simplify the right side.

Divide $14$ by $7$ .

$x^{2} - \frac{9 x}{7} = 2$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{9}{14})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{9 x}{7} + {(- \frac{9}{14})}^{2} = 2 + {(- \frac{9}{14})}^{2}$

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 $- \frac{9}{14}$ .

$x^{2} - \frac{9 x}{7} + {(- 1)}^{2} {(\frac{9}{14})}^{2} = 2 + {(- \frac{9}{14})}^{2}$

Apply the product rule to $\frac{9}{14}$ .

$x^{2} - \frac{9 x}{7} + {(- 1)}^{2} \frac{9^{2}}{14^{2}} = 2 + {(- \frac{9}{14})}^{2}$

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

$x^{2} - \frac{9 x}{7} + 1 \frac{9^{2}}{14^{2}} = 2 + {(- \frac{9}{14})}^{2}$

Multiply $\frac{9^{2}}{14^{2}}$ by $1$ .

$x^{2} - \frac{9 x}{7} + \frac{9^{2}}{14^{2}} = 2 + {(- \frac{9}{14})}^{2}$

Raise $9$ to the power of $2$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{14^{2}} = 2 + {(- \frac{9}{14})}^{2}$

Raise $14$ to the power of $2$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + {(- \frac{9}{14})}^{2}$

Simplify the right side.

Simplify $2 + {(- \frac{9}{14})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{9}{14}$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + {(- 1)}^{2} {(\frac{9}{14})}^{2}$

Apply the product rule to $\frac{9}{14}$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + {(- 1)}^{2} \frac{9^{2}}{14^{2}}$

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

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + 1 \frac{9^{2}}{14^{2}}$

Multiply $\frac{9^{2}}{14^{2}}$ by $1$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + \frac{9^{2}}{14^{2}}$

Raise $9$ to the power of $2$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + \frac{81}{14^{2}}$

Raise $14$ to the power of $2$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 + \frac{81}{196}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{196}{196}$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = 2 \cdot \frac{196}{196} + \frac{81}{196}$

Combine $2$ and $\frac{196}{196}$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = \frac{2 \cdot 196}{196} + \frac{81}{196}$

Combine the numerators over the common denominator.

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = \frac{2 \cdot 196 + 81}{196}$

Simplify the numerator.

Multiply $2$ by $196$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = \frac{392 + 81}{196}$

Add $392$ and $81$ .

$x^{2} - \frac{9 x}{7} + \frac{81}{196} = \frac{473}{196}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{9}{14})}^{2}$ .

${(x - \frac{9}{14})}^{2} = \frac{473}{196}$

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.

$x - \frac{9}{14} = \pm \sqrt{\frac{473}{196}}$

Simplify $\pm \sqrt{\frac{473}{196}}$ .

Rewrite $\sqrt{\frac{473}{196}}$ as $\frac{\sqrt{473}}{\sqrt{196}}$ .

$x - \frac{9}{14} = \pm \frac{\sqrt{473}}{\sqrt{196}}$

Simplify the denominator.

Rewrite $196$ as $14^{2}$ .

$x - \frac{9}{14} = \pm \frac{\sqrt{473}}{\sqrt{14^{2}}}$

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

$x - \frac{9}{14} = \pm \frac{\sqrt{473}}{14}$

Add $\frac{9}{14}$ to both sides of the equation.

$x = \pm \frac{\sqrt{473}}{14} + \frac{9}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{473}}{14} + \frac{9}{14}$

Decimal Form:

$x = 2.19632594 \dots, - 0.91061165 \dots$

Solve by Completing the Square 7x^2-9x=14

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