Solve by Completing the Square x^2-5x=-9/4

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine fractions.

Combine the numerators over the common denominator.

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

Simplify the expression.

Add $- 9$ and $25$ .

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

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 $x$ .

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

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

Simplify $\pm \sqrt{4}$ .

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

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

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

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

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.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Add $4$ and $5$ .

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 2$ by $2$ .

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

Add $- 4$ and $5$ .

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

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

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

Solve by Completing the Square x^2-5x=-9/4

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