Solve by Completing the Square x^2-11x+24=0

$x^{2} - 11 x + 24 = 0$

Step 1

Subtract $24$ from both sides of the equation.

$x^{2} - 11 x = - 24$

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 $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} - 11 x + {(- \frac{11}{2})}^{2} = - 24 + {(- \frac{11}{2})}^{2}$

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 $- \frac{11}{2}$ .

$x^{2} - 11 x + {(- 1)}^{2} {(\frac{11}{2})}^{2} = - 24 + {(- \frac{11}{2})}^{2}$

Apply the product rule to $\frac{11}{2}$ .

$x^{2} - 11 x + {(- 1)}^{2} \frac{11^{2}}{2^{2}} = - 24 + {(- \frac{11}{2})}^{2}$

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

$x^{2} - 11 x + 1 \frac{11^{2}}{2^{2}} = - 24 + {(- \frac{11}{2})}^{2}$

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

$x^{2} - 11 x + \frac{11^{2}}{2^{2}} = - 24 + {(- \frac{11}{2})}^{2}$

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

$x^{2} - 11 x + \frac{121}{2^{2}} = - 24 + {(- \frac{11}{2})}^{2}$

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

$x^{2} - 11 x + \frac{121}{4} = - 24 + {(- \frac{11}{2})}^{2}$

Simplify the right side.

Simplify $- 24 + {(- \frac{11}{2})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{11}{2}$ .

$x^{2} - 11 x + \frac{121}{4} = - 24 + {(- 1)}^{2} {(\frac{11}{2})}^{2}$

Apply the product rule to $\frac{11}{2}$ .

$x^{2} - 11 x + \frac{121}{4} = - 24 + {(- 1)}^{2} \frac{11^{2}}{2^{2}}$

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

$x^{2} - 11 x + \frac{121}{4} = - 24 + 1 \frac{11^{2}}{2^{2}}$

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

$x^{2} - 11 x + \frac{121}{4} = - 24 + \frac{11^{2}}{2^{2}}$

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

$x^{2} - 11 x + \frac{121}{4} = - 24 + \frac{121}{2^{2}}$

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

$x^{2} - 11 x + \frac{121}{4} = - 24 + \frac{121}{4}$

To write $- 24$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 11 x + \frac{121}{4} = - 24 \cdot \frac{4}{4} + \frac{121}{4}$

Combine $- 24$ and $\frac{4}{4}$ .

$x^{2} - 11 x + \frac{121}{4} = \frac{- 24 \cdot 4}{4} + \frac{121}{4}$

Combine the numerators over the common denominator.

$x^{2} - 11 x + \frac{121}{4} = \frac{- 24 \cdot 4 + 121}{4}$

Simplify the numerator.

Multiply $- 24$ by $4$ .

$x^{2} - 11 x + \frac{121}{4} = \frac{- 96 + 121}{4}$

Add $- 96$ and $121$ .

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

Step 5

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

${(x - \frac{11}{2})}^{2} = \frac{25}{4}$

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.

$x - \frac{11}{2} = \pm \sqrt{\frac{25}{4}}$

Simplify $\pm \sqrt{\frac{25}{4}}$ .

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$x - \frac{11}{2} = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$x - \frac{11}{2} = \pm \frac{\sqrt{5^{2}}}{\sqrt{4}}$

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

$x - \frac{11}{2} = \pm \frac{5}{\sqrt{4}}$

Simplify the denominator.

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

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

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

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

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.

$x - \frac{11}{2} = \frac{5}{2}$

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

Add $\frac{11}{2}$ to both sides of the equation.

$x = \frac{5}{2} + \frac{11}{2}$

Combine the numerators over the common denominator.

$x = \frac{5 + 11}{2}$

Add $5$ and $11$ .

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

Divide $16$ by $2$ .

$x = 8$

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

$x - \frac{11}{2} = - \frac{5}{2}$

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

Add $\frac{11}{2}$ to both sides of the equation.

$x = - \frac{5}{2} + \frac{11}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 5 + 11}{2}$

Add $- 5$ and $11$ .

$x = \frac{6}{2}$

Divide $6$ by $2$ .

$x = 3$

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

$x = 8, 3$

Solve by Completing the Square x^2-11x+24=0

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