Algebra Examples

$x^{2} - 5 x = - \frac{9}{4}$

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.

$x^{2} - 5 x + {(- \frac{5}{2})}^{2} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

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 $- \frac{5}{2}$ .

$x^{2} - 5 x + {(- 1)}^{2} {(\frac{5}{2})}^{2} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.1.1.1.2

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

$x^{2} - 5 x + {(- 1)}^{2} \frac{5^{2}}{2^{2}} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.1.1.2

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

$x^{2} - 5 x + 1 \frac{5^{2}}{2^{2}} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.1.1.3

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

$x^{2} - 5 x + \frac{5^{2}}{2^{2}} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.1.1.4

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

$x^{2} - 5 x + \frac{25}{2^{2}} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.1.1.5

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + {(- \frac{5}{2})}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{9}{4} + {(- \frac{5}{2})}^{2}$ .

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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 $- \frac{5}{2}$ .

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + {(- 1)}^{2} {(\frac{5}{2})}^{2}$

Step 3.2.1.1.1.2

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + {(- 1)}^{2} \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.2

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + 1 \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.3

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + \frac{5^{2}}{2^{2}}$

Step 3.2.1.1.4

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + \frac{25}{2^{2}}$

Step 3.2.1.1.5

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

$x^{2} - 5 x + \frac{25}{4} = - \frac{9}{4} + \frac{25}{4}$

Step 3.2.1.2

Combine fractions.

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Step 3.2.1.2.1

Combine the numerators over the common denominator.

$x^{2} - 5 x + \frac{25}{4} = \frac{- 9 + 25}{4}$

Step 3.2.1.2.2

Simplify the expression.

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Step 3.2.1.2.2.1

Add $- 9$ and $25$ .

$x^{2} - 5 x + \frac{25}{4} = \frac{16}{4}$

Step 3.2.1.2.2.2

Divide $16$ by $4$ .

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

Step 4

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

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

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.

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

Step 5.2

Simplify $\pm \sqrt{4}$ .

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

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

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Step 5.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.2.4

Combine the numerators over the common denominator.

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

Step 5.3.2.5

Simplify the numerator.

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Step 5.3.2.5.1

Multiply $2$ by $2$ .

$x = \frac{4 + 5}{2}$

Step 5.3.2.5.2

Add $4$ and $5$ .

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

Step 5.3.3

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

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

Step 5.3.4

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

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Step 5.3.4.1

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

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

Step 5.3.4.2

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

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

Step 5.3.4.3

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

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

Step 5.3.4.4

Combine the numerators over the common denominator.

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

Step 5.3.4.5

Simplify the numerator.

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Step 5.3.4.5.1

Multiply $- 2$ by $2$ .

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

Step 5.3.4.5.2

Add $- 4$ and $5$ .

$x = \frac{1}{2}$

Step 5.3.5

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

$x = \frac{9}{2}, \frac{1}{2}$