Finite Math Examples

$x^{2} - 5 x = 5$

Step 1

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{5}{2})}^{2}$

Step 2

Add the term to each side of the equation.

$x^{2} - 5 x + {(- \frac{5}{2})}^{2} = 5 + {(- \frac{5}{2})}^{2}$

Step 3

Simplify the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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Step 3.1.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$x^{2} - 5 x + {(- 1)}^{2} {(\frac{5}{2})}^{2} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.1.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$x^{2} - 5 x + {(- 1)}^{2} \frac{5^{2}}{2^{2}} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.1.1.2

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

$x^{2} - 5 x + 1 \frac{5^{2}}{2^{2}} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.1.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$x^{2} - 5 x + \frac{5^{2}}{2^{2}} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.1.1.4

Raise $5$ to the power of $2$ .

$x^{2} - 5 x + \frac{25}{2^{2}} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.1.1.5

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

$x^{2} - 5 x + \frac{25}{4} = 5 + {(- \frac{5}{2})}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $5 + {(- \frac{5}{2})}^{2}$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$x^{2} - 5 x + \frac{25}{4} = 5 + {(- 1)}^{2} {(\frac{5}{2})}^{2}$

Step 3.2.1.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$x^{2} - 5 x + \frac{25}{4} = 5 + {(- 1)}^{2} \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.2

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

$x^{2} - 5 x + \frac{25}{4} = 5 + 1 \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$x^{2} - 5 x + \frac{25}{4} = 5 + \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.4

Raise $5$ to the power of $2$ .

$x^{2} - 5 x + \frac{25}{4} = 5 + \frac{25}{2^{2}}$

Step 3.2.1.1.5

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

$x^{2} - 5 x + \frac{25}{4} = 5 + \frac{25}{4}$

Step 3.2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 5 x + \frac{25}{4} = 5 \cdot \frac{4}{4} + \frac{25}{4}$

Step 3.2.1.3

Combine $5$ and $\frac{4}{4}$ .

$x^{2} - 5 x + \frac{25}{4} = \frac{5 \cdot 4}{4} + \frac{25}{4}$

Step 3.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 5 x + \frac{25}{4} = \frac{5 \cdot 4 + 25}{4}$

Step 3.2.1.5

Simplify the numerator.

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Step 3.2.1.5.1

Multiply $5$ by $4$ .

$x^{2} - 5 x + \frac{25}{4} = \frac{20 + 25}{4}$

Step 3.2.1.5.2

Add $20$ and $25$ .

$x^{2} - 5 x + \frac{25}{4} = \frac{45}{4}$

Step 4

Factor the perfect trinomial square into ${(x - \frac{5}{2})}^{2}$ .

${(x - \frac{5}{2})}^{2} = \frac{45}{4}$

Step 5

Solve the equation for $x$ .

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Step 5.1

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

$x - \frac{5}{2} = \pm \sqrt{\frac{45}{4}}$

Step 5.2

Simplify $\pm \sqrt{\frac{45}{4}}$ .

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Step 5.2.1

Rewrite $\sqrt{\frac{45}{4}}$ as $\frac{\sqrt{45}}{\sqrt{4}}$ .

$x - \frac{5}{2} = \pm \frac{\sqrt{45}}{\sqrt{4}}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 5.2.2.1.1

Factor $9$ out of $45$ .

$x - \frac{5}{2} = \pm \frac{\sqrt{9 (5)}}{\sqrt{4}}$

Step 5.2.2.1.2

Rewrite $9$ as $3^{2}$ .

$x - \frac{5}{2} = \pm \frac{\sqrt{3^{2} \cdot 5}}{\sqrt{4}}$

Step 5.2.2.2

Pull terms out from under the radical.

$x - \frac{5}{2} = \pm \frac{3 \sqrt{5}}{\sqrt{4}}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

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

$x - \frac{5}{2} = \pm \frac{3 \sqrt{5}}{\sqrt{2^{2}}}$

Step 5.2.3.2

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

$x - \frac{5}{2} = \pm \frac{3 \sqrt{5}}{2}$

Step 5.3

Add $\frac{5}{2}$ to both sides of the equation.

$x = \pm \frac{3 \sqrt{5}}{2} + \frac{5}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{3 \sqrt{5}}{2} + \frac{5}{2}$

Decimal Form:

$x = 5.85410196 \dots, - 0.85410196 \dots$