Solve by Completing the Square x^2+5x-1=0

$x^{2} + 5 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$x^{2} + 5 x = 1$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{2})}^{2}$

Step 3

Add the term to each side of the equation.

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

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

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

Raise $5$ to the power of $2$ .

$x^{2} + 5 x + \frac{25}{2^{2}} = 1 + {(\frac{5}{2})}^{2}$

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

$x^{2} + 5 x + \frac{25}{4} = 1 + {(\frac{5}{2})}^{2}$

Simplify the right side.

Simplify $1 + {(\frac{5}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{5^{2}}{2^{2}}$

Raise $5$ to the power of $2$ .

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{25}{2^{2}}$

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

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{25}{4}$

Simplify the expression.

Write $1$ as a fraction with a common denominator.

$x^{2} + 5 x + \frac{25}{4} = \frac{4}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$x^{2} + 5 x + \frac{25}{4} = \frac{4 + 25}{4}$

Add $4$ and $25$ .

$x^{2} + 5 x + \frac{25}{4} = \frac{29}{4}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{5}{2})}^{2}$ .

${(x + \frac{5}{2})}^{2} = \frac{29}{4}$

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.

$x + \frac{5}{2} = \pm \sqrt{\frac{29}{4}}$

Simplify $\pm \sqrt{\frac{29}{4}}$ .

Rewrite $\sqrt{\frac{29}{4}}$ as $\frac{\sqrt{29}}{\sqrt{4}}$ .

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{\sqrt{4}}$

Simplify the denominator.

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

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{\sqrt{2^{2}}}$

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

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{2}$

Subtract $\frac{5}{2}$ from both sides of the equation.

$x = \pm \frac{\sqrt{29}}{2} - \frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{29}}{2} - \frac{5}{2}$

Decimal Form:

$x = 0.19258240 \dots, - 5.19258240 \dots$

Solve by Completing the Square x^2+5x-1=0

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