Finite Math Examples

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

Step 1

Add $24$ to 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.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

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

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Step 4.1.1.1.1

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

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

Step 4.1.1.1.2

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

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

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

Step 4.1.1.5

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

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

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

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Step 4.2.1.1.1.1

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

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

Step 4.2.1.1.1.2

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.1.3

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

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

Step 4.2.1.1.4

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

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

Step 4.2.1.1.5

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Combine the numerators over the common denominator.

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $24$ by $4$ .

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

Step 4.2.1.5.2

Add $96$ and $121$ .

$x^{2} - 11 x + \frac{121}{4} = \frac{217}{4}$

Step 5

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

${(x - \frac{11}{2})}^{2} = \frac{217}{4}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x - \frac{11}{2} = \pm \sqrt{\frac{217}{4}}$

Step 6.2

Simplify $\pm \sqrt{\frac{217}{4}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{217}{4}}$ as $\frac{\sqrt{217}}{\sqrt{4}}$ .

$x - \frac{11}{2} = \pm \frac{\sqrt{217}}{\sqrt{4}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$x - \frac{11}{2} = \pm \frac{\sqrt{217}}{\sqrt{2^{2}}}$

Step 6.2.2.2

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

$x - \frac{11}{2} = \pm \frac{\sqrt{217}}{2}$

Step 6.3

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

$x = \pm \frac{\sqrt{217}}{2} + \frac{11}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{217}}{2} + \frac{11}{2}$

Decimal Form:

$x = 12.86545993 \dots, - 1.86545993 \dots$