Solve by Completing the Square 2x^2+9x+1=0

$2 x^{2} + 9 x + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$2 x^{2} + 9 x = - 1$

Step 2

Divide each term in $2 x^{2} + 9 x = - 1$ by $2$ and simplify.

Divide each term in $2 x^{2} + 9 x = - 1$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{9 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{9 x}{2} = \frac{- 1}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{9 x}{2} = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} + \frac{9 x}{2} = - \frac{1}{2}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{9}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{9 x}{2} + {(\frac{9}{4})}^{2} = - \frac{1}{2} + {(\frac{9}{4})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{9}{4}$ .

$x^{2} + \frac{9 x}{2} + \frac{9^{2}}{4^{2}} = - \frac{1}{2} + {(\frac{9}{4})}^{2}$

Raise $9$ to the power of $2$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{4^{2}} = - \frac{1}{2} + {(\frac{9}{4})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{1}{2} + {(\frac{9}{4})}^{2}$

Simplify the right side.

Simplify $- \frac{1}{2} + {(\frac{9}{4})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{9}{4}$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{1}{2} + \frac{9^{2}}{4^{2}}$

Raise $9$ to the power of $2$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{1}{2} + \frac{81}{4^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{1}{2} + \frac{81}{16}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{1}{2} \cdot \frac{8}{8} + \frac{81}{16}$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{8}{2 \cdot 8} + \frac{81}{16}$

Multiply $2$ by $8$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = - \frac{8}{16} + \frac{81}{16}$

Combine the numerators over the common denominator.

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = \frac{- 8 + 81}{16}$

Add $- 8$ and $81$ .

$x^{2} + \frac{9 x}{2} + \frac{81}{16} = \frac{73}{16}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{9}{4})}^{2}$ .

${(x + \frac{9}{4})}^{2} = \frac{73}{16}$

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.

$x + \frac{9}{4} = \pm \sqrt{\frac{73}{16}}$

Simplify $\pm \sqrt{\frac{73}{16}}$ .

Rewrite $\sqrt{\frac{73}{16}}$ as $\frac{\sqrt{73}}{\sqrt{16}}$ .

$x + \frac{9}{4} = \pm \frac{\sqrt{73}}{\sqrt{16}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$x + \frac{9}{4} = \pm \frac{\sqrt{73}}{\sqrt{4^{2}}}$

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

$x + \frac{9}{4} = \pm \frac{\sqrt{73}}{4}$

Subtract $\frac{9}{4}$ from both sides of the equation.

$x = \pm \frac{\sqrt{73}}{4} - \frac{9}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{73}}{4} - \frac{9}{4}$

Decimal Form:

$x = - 0.11399906 \dots, - 4.38600093 \dots$

Solve by Completing the Square 2x^2+9x+1=0

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