Solve by Completing the Square (x^2)/2+7/2x=1

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

Step 1

Combine $\frac{7}{2}$ and $x$ .

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

Step 2

Multiply each term in $\frac{x^{2}}{2} + \frac{7 x}{2} = 1$ by $2$ to eliminate the fractions.

Multiply each term in $\frac{x^{2}}{2} + \frac{7 x}{2} = 1$ by $2$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{2} + 7 x = 1 \cdot 2$

Simplify the right side.

Multiply $2$ by $1$ .

$x^{2} + 7 x = 2$

Step 3

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 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

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

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

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

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

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

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $2$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $4$ .

$x^{2} + 7 x + \frac{49}{4} = \frac{8 + 49}{4}$

Add $8$ and $49$ .

$x^{2} + 7 x + \frac{49}{4} = \frac{57}{4}$

Step 6

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

${(x + \frac{7}{2})}^{2} = \frac{57}{4}$

Step 7

Solve the equation for $x$ .

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

$x + \frac{7}{2} = \pm \sqrt{\frac{57}{4}}$

Simplify $\pm \sqrt{\frac{57}{4}}$ .

Rewrite $\sqrt{\frac{57}{4}}$ as $\frac{\sqrt{57}}{\sqrt{4}}$ .

$x + \frac{7}{2} = \pm \frac{\sqrt{57}}{\sqrt{4}}$

Simplify the denominator.

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

$x + \frac{7}{2} = \pm \frac{\sqrt{57}}{\sqrt{2^{2}}}$

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

$x + \frac{7}{2} = \pm \frac{\sqrt{57}}{2}$

Subtract $\frac{7}{2}$ from both sides of the equation.

$x = \pm \frac{\sqrt{57}}{2} - \frac{7}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{57}}{2} - \frac{7}{2}$

Decimal Form:

$x = 0.27491721 \dots, - 7.27491721 \dots$

Solve by Completing the Square (x^2)/2+7/2x=1

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