Precalculus Examples

$x^{2} - 7 x + \frac{5}{4} = 0$

Step 1

Subtract $\frac{5}{4}$ from both sides of the equation.

$x^{2} - 7 x = - \frac{5}{4}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{7}{2})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 7 x + {(- \frac{7}{2})}^{2} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

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 $- \frac{7}{2}$ .

$x^{2} - 7 x + {(- 1)}^{2} {(\frac{7}{2})}^{2} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.1.1.1.2

Apply the product rule to $\frac{7}{2}$ .

$x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.1.1.2

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

$x^{2} - 7 x + 1 \frac{7^{2}}{2^{2}} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.1.1.3

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$x^{2} - 7 x + \frac{7^{2}}{2^{2}} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.1.1.4

Raise $7$ to the power of $2$ .

$x^{2} - 7 x + \frac{49}{2^{2}} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.1.1.5

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

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + {(- \frac{7}{2})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{4} + {(- \frac{7}{2})}^{2}$ .

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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 $- \frac{7}{2}$ .

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + {(- 1)}^{2} {(\frac{7}{2})}^{2}$

Step 4.2.1.1.1.2

Apply the product rule to $\frac{7}{2}$ .

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + {(- 1)}^{2} \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.2

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

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + 1 \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.3

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.4

Raise $7$ to the power of $2$ .

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + \frac{49}{2^{2}}$

Step 4.2.1.1.5

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

$x^{2} - 7 x + \frac{49}{4} = - \frac{5}{4} + \frac{49}{4}$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x^{2} - 7 x + \frac{49}{4} = \frac{- 5 + 49}{4}$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Add $- 5$ and $49$ .

$x^{2} - 7 x + \frac{49}{4} = \frac{44}{4}$

Step 4.2.1.2.2.2

Divide $44$ by $4$ .

$x^{2} - 7 x + \frac{49}{4} = 11$

Step 5

Factor the perfect trinomial square into ${(x - \frac{7}{2})}^{2}$ .

${(x - \frac{7}{2})}^{2} = 11$

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.

$x - \frac{7}{2} = \pm \sqrt{11}$

Step 6.2

Add $\frac{7}{2}$ to both sides of the equation.

$x = \pm \sqrt{11} + \frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{11} + \frac{7}{2}$

Decimal Form:

$x = 6.81662479 \dots, 0.18337520 \dots$