Solve by Completing the Square x^2+x=7/4

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{2})}^{2}$

Step 2

Add the term to each side of the equation.

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

Step 3

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Simplify the right side.

Simplify $\frac{7}{4} + {(\frac{1}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Combine fractions.

Combine the numerators over the common denominator.

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

Simplify the expression.

Add $7$ and $1$ .

$x^{2} + x + \frac{1}{4} = \frac{8}{4}$

Divide $8$ by $4$ .

$x^{2} + x + \frac{1}{4} = 2$

Step 4

Factor the perfect trinomial square into ${(x + \frac{1}{2})}^{2}$ .

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

Step 5

Solve the equation for $x$ .

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

$x + \frac{1}{2} = \pm \sqrt{2}$

Subtract $\frac{1}{2}$ from both sides of the equation.

$x = \pm \sqrt{2} - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{2} - \frac{1}{2}$

Decimal Form:

$x = 0.91421356 \dots, - 1.91421356 \dots$

Solve by Completing the Square x^2+x=7/4

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