Solve by Completing the Square z(z+9)=1

$z (z + 9) = 1$

Simplify $z (z + 9)$ .

Apply the distributive property.

$z \cdot z + z \cdot 9 = 1$

Simplify the expression.

Multiply $z$ by $z$ .

$z^{2} + z \cdot 9 = 1$

Move $9$ to the left of $z$ .

$z^{2} + 9 z = 1$

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

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

Add the term to each side of the equation.

$z^{2} + 9 z + {(\frac{9}{2})}^{2} = 1 + {(\frac{9}{2})}^{2}$

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{9}{2}$ .

$z^{2} + 9 z + \frac{9^{2}}{2^{2}} = 1 + {(\frac{9}{2})}^{2}$

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

$z^{2} + 9 z + \frac{81}{2^{2}} = 1 + {(\frac{9}{2})}^{2}$

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

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

Simplify the right side.

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

Simplify each term.

Apply the product rule to $\frac{9}{2}$ .

$z^{2} + 9 z + \frac{81}{4} = 1 + \frac{9^{2}}{2^{2}}$

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

$z^{2} + 9 z + \frac{81}{4} = 1 + \frac{81}{2^{2}}$

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

$z^{2} + 9 z + \frac{81}{4} = 1 + \frac{81}{4}$

Simplify the expression.

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

$z^{2} + 9 z + \frac{81}{4} = \frac{4}{4} + \frac{81}{4}$

Combine the numerators over the common denominator.

$z^{2} + 9 z + \frac{81}{4} = \frac{4 + 81}{4}$

Add $4$ and $81$ .

$z^{2} + 9 z + \frac{81}{4} = \frac{85}{4}$

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

${(z + \frac{9}{2})}^{2} = \frac{85}{4}$

Solve the equation for $z$ .

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

$z + \frac{9}{2} = \pm \sqrt{\frac{85}{4}}$

Simplify $\pm \sqrt{\frac{85}{4}}$ .

Rewrite $\sqrt{\frac{85}{4}}$ as $\frac{\sqrt{85}}{\sqrt{4}}$ .

$z + \frac{9}{2} = \pm \frac{\sqrt{85}}{\sqrt{4}}$

Simplify the denominator.

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

$z + \frac{9}{2} = \pm \frac{\sqrt{85}}{\sqrt{2^{2}}}$

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

$z + \frac{9}{2} = \pm \frac{\sqrt{85}}{2}$

Subtract $\frac{9}{2}$ from both sides of the equation.

$z = \pm \frac{\sqrt{85}}{2} - \frac{9}{2}$

The result can be shown in multiple forms.

Exact Form:

$z = \pm \frac{\sqrt{85}}{2} - \frac{9}{2}$

Decimal Form:

$z = 0.10977222 \dots, - 9.10977222 \dots$

Solve by Completing the Square z(z+9)=1

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