Solve by Completing the Square x^2-3/4x=5

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

Step 1

Simplify each term.

Combine $x$ and $\frac{3}{4}$ .

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

Move $3$ to the left of $x$ .

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{8})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{3 x}{4} + {(- \frac{3}{8})}^{2} = 5 + {(- \frac{3}{8})}^{2}$

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 $- \frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + {(- 1)}^{2} {(\frac{3}{8})}^{2} = 5 + {(- \frac{3}{8})}^{2}$

Apply the product rule to $\frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + {(- 1)}^{2} \frac{3^{2}}{8^{2}} = 5 + {(- \frac{3}{8})}^{2}$

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

$x^{2} - \frac{3 x}{4} + 1 \frac{3^{2}}{8^{2}} = 5 + {(- \frac{3}{8})}^{2}$

Multiply $\frac{3^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{4} + \frac{3^{2}}{8^{2}} = 5 + {(- \frac{3}{8})}^{2}$

Raise $3$ to the power of $2$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{8^{2}} = 5 + {(- \frac{3}{8})}^{2}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + {(- \frac{3}{8})}^{2}$

Simplify the right side.

Simplify $5 + {(- \frac{3}{8})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + {(- 1)}^{2} {(\frac{3}{8})}^{2}$

Apply the product rule to $\frac{3}{8}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + {(- 1)}^{2} \frac{3^{2}}{8^{2}}$

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

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + 1 \frac{3^{2}}{8^{2}}$

Multiply $\frac{3^{2}}{8^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + \frac{3^{2}}{8^{2}}$

Raise $3$ to the power of $2$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + \frac{9}{8^{2}}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 + \frac{9}{64}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = 5 \cdot \frac{64}{64} + \frac{9}{64}$

Combine $5$ and $\frac{64}{64}$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{5 \cdot 64}{64} + \frac{9}{64}$

Combine the numerators over the common denominator.

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{5 \cdot 64 + 9}{64}$

Simplify the numerator.

Multiply $5$ by $64$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{320 + 9}{64}$

Add $320$ and $9$ .

$x^{2} - \frac{3 x}{4} + \frac{9}{64} = \frac{329}{64}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{3}{8})}^{2}$ .

${(x - \frac{3}{8})}^{2} = \frac{329}{64}$

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.

$x - \frac{3}{8} = \pm \sqrt{\frac{329}{64}}$

Simplify $\pm \sqrt{\frac{329}{64}}$ .

Rewrite $\sqrt{\frac{329}{64}}$ as $\frac{\sqrt{329}}{\sqrt{64}}$ .

$x - \frac{3}{8} = \pm \frac{\sqrt{329}}{\sqrt{64}}$

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$x - \frac{3}{8} = \pm \frac{\sqrt{329}}{\sqrt{8^{2}}}$

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

$x - \frac{3}{8} = \pm \frac{\sqrt{329}}{8}$

Add $\frac{3}{8}$ to both sides of the equation.

$x = \pm \frac{\sqrt{329}}{8} + \frac{3}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{329}}{8} + \frac{3}{8}$

Decimal Form:

$x = 2.64229464 \dots, - 1.89229464 \dots$

Solve by Completing the Square x^2-3/4x=5

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