Algebra Examples

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

Step 1

Subtract $\frac{7}{4}$ from both sides of the equation.

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

Step 2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 1$

$c = - \frac{7}{4} = 0$

Step 3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = 0$

Step 4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 1}{2 \cdot 1}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 4.2.1.1

Rewrite $- 1$ as $- 1 (1)$ .

$d = \frac{- 1 (1)}{2 \cdot 1}$

Step 4.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1}{2 \cdot 1}$

Step 4.2.1.3

Rewrite the expression.

$d = \frac{- 1}{2}$

Step 4.2.2

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Step 5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - \frac{7}{4} - \frac{{(- 1)}^{2}}{4 \cdot 1}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$e = - \frac{7}{4} - \frac{1}{4 \cdot 1}$

Step 5.2.1.2

Multiply $4$ by $1$ .

$e = - \frac{7}{4} - \frac{1}{4}$

Step 5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 7 - 1}{4}$

Step 5.2.3

Subtract $1$ from $- 7$ .

$e = \frac{- 8}{4}$

Step 5.2.4

Divide $- 8$ by $4$ .

$e = - 2$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{1}{2})}^{2} - 2$ .

${(x - \frac{1}{2})}^{2} - 2 = 0$