Solve by Completing the Square 2x^2-14x+10=26

$2 x^{2} - 14 x + 10 = 26$

Step 1

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

Subtract $10$ from both sides of the equation.

$2 x^{2} - 14 x = 26 - 10$

Subtract $10$ from $26$ .

$2 x^{2} - 14 x = 16$

Step 2

Divide each term in $2 x^{2} - 14 x = 16$ by $2$ and simplify.

Divide each term in $2 x^{2} - 14 x = 16$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{- 14 x}{2} = \frac{16}{2}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{- 14 x}{2} = \frac{16}{2}$

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

$x^{2} + \frac{- 14 x}{2} = \frac{16}{2}$

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14 x$ .

$x^{2} + \frac{2 (- 7 x)}{2} = \frac{16}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (- 7 x)}{2 (1)} = \frac{16}{2}$

Cancel the common factor.

$x^{2} + \frac{2 (- 7 x)}{2 \cdot 1} = \frac{16}{2}$

Rewrite the expression.

$x^{2} + \frac{- 7 x}{1} = \frac{16}{2}$

Divide $- 7 x$ by $1$ .

$x^{2} - 7 x = \frac{16}{2}$

Simplify the right side.

Divide $16$ by $2$ .

$x^{2} - 7 x = 8$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} - 7 x + {(- \frac{7}{2})}^{2} = 8 + {(- \frac{7}{2})}^{2}$

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 $- \frac{7}{2}$ .

$x^{2} - 7 x + {(- 1)}^{2} {(\frac{7}{2})}^{2} = 8 + {(- \frac{7}{2})}^{2}$

Apply the product rule to $\frac{7}{2}$ .

$x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = 8 + {(- \frac{7}{2})}^{2}$

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

$x^{2} - 7 x + 1 \frac{7^{2}}{2^{2}} = 8 + {(- \frac{7}{2})}^{2}$

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

$x^{2} - 7 x + \frac{7^{2}}{2^{2}} = 8 + {(- \frac{7}{2})}^{2}$

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

$x^{2} - 7 x + \frac{49}{2^{2}} = 8 + {(- \frac{7}{2})}^{2}$

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

$x^{2} - 7 x + \frac{49}{4} = 8 + {(- \frac{7}{2})}^{2}$

Simplify the right side.

Simplify $8 + {(- \frac{7}{2})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{7}{2}$ .

$x^{2} - 7 x + \frac{49}{4} = 8 + {(- 1)}^{2} {(\frac{7}{2})}^{2}$

Apply the product rule to $\frac{7}{2}$ .

$x^{2} - 7 x + \frac{49}{4} = 8 + {(- 1)}^{2} \frac{7^{2}}{2^{2}}$

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

$x^{2} - 7 x + \frac{49}{4} = 8 + 1 \frac{7^{2}}{2^{2}}$

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

$x^{2} - 7 x + \frac{49}{4} = 8 + \frac{7^{2}}{2^{2}}$

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

$x^{2} - 7 x + \frac{49}{4} = 8 + \frac{49}{2^{2}}$

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

$x^{2} - 7 x + \frac{49}{4} = 8 + \frac{49}{4}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 7 x + \frac{49}{4} = 8 \cdot \frac{4}{4} + \frac{49}{4}$

Combine $8$ and $\frac{4}{4}$ .

$x^{2} - 7 x + \frac{49}{4} = \frac{8 \cdot 4}{4} + \frac{49}{4}$

Combine the numerators over the common denominator.

$x^{2} - 7 x + \frac{49}{4} = \frac{8 \cdot 4 + 49}{4}$

Simplify the numerator.

Multiply $8$ by $4$ .

$x^{2} - 7 x + \frac{49}{4} = \frac{32 + 49}{4}$

Add $32$ and $49$ .

$x^{2} - 7 x + \frac{49}{4} = \frac{81}{4}$

Step 6

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

${(x - \frac{7}{2})}^{2} = \frac{81}{4}$

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.

$x - \frac{7}{2} = \pm \sqrt{\frac{81}{4}}$

Simplify $\pm \sqrt{\frac{81}{4}}$ .

Rewrite $\sqrt{\frac{81}{4}}$ as $\frac{\sqrt{81}}{\sqrt{4}}$ .

$x - \frac{7}{2} = \pm \frac{\sqrt{81}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $81$ as $9^{2}$ .

$x - \frac{7}{2} = \pm \frac{\sqrt{9^{2}}}{\sqrt{4}}$

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

$x - \frac{7}{2} = \pm \frac{9}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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

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

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.

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

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

Add $\frac{7}{2}$ to both sides of the equation.

$x = \frac{9}{2} + \frac{7}{2}$

Combine the numerators over the common denominator.

$x = \frac{9 + 7}{2}$

Add $9$ and $7$ .

$x = \frac{16}{2}$

Divide $16$ by $2$ .

$x = 8$

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

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

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

Add $\frac{7}{2}$ to both sides of the equation.

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

Combine the numerators over the common denominator.

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

Add $- 9$ and $7$ .

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

Divide $- 2$ by $2$ .

$x = - 1$

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

$x = 8, - 1$

Solve by Completing the Square 2x^2-14x+10=26

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