Solve by Completing the Square x^2+3x-9=0

$x^{2} + 3 x - 9 = 0$

Step 1

Add $9$ to both sides of the equation.

$x^{2} + 3 x = 9$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{3}{2})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 3 x + {(\frac{3}{2})}^{2} = 9 + {(\frac{3}{2})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{3^{2}}{2^{2}} = 9 + {(\frac{3}{2})}^{2}$

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{2^{2}} = 9 + {(\frac{3}{2})}^{2}$

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

$x^{2} + 3 x + \frac{9}{4} = 9 + {(\frac{3}{2})}^{2}$

Simplify the right side.

Simplify $9 + {(\frac{3}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{9}{4} = 9 + \frac{3^{2}}{2^{2}}$

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{4} = 9 + \frac{9}{2^{2}}$

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

$x^{2} + 3 x + \frac{9}{4} = 9 + \frac{9}{4}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + 3 x + \frac{9}{4} = 9 \cdot \frac{4}{4} + \frac{9}{4}$

Combine $9$ and $\frac{4}{4}$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{9 \cdot 4}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

$x^{2} + 3 x + \frac{9}{4} = \frac{9 \cdot 4 + 9}{4}$

Simplify the numerator.

Multiply $9$ by $4$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{36 + 9}{4}$

Add $36$ and $9$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{45}{4}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{3}{2})}^{2}$ .

${(x + \frac{3}{2})}^{2} = \frac{45}{4}$

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.

$x + \frac{3}{2} = \pm \sqrt{\frac{45}{4}}$

Simplify $\pm \sqrt{\frac{45}{4}}$ .

Rewrite $\sqrt{\frac{45}{4}}$ as $\frac{\sqrt{45}}{\sqrt{4}}$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{45}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{9 (5)}}{\sqrt{4}}$

Rewrite $9$ as $3^{2}$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{3^{2} \cdot 5}}{\sqrt{4}}$

Pull terms out from under the radical.

$x + \frac{3}{2} = \pm \frac{3 \sqrt{5}}{\sqrt{4}}$

Simplify the denominator.

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

$x + \frac{3}{2} = \pm \frac{3 \sqrt{5}}{\sqrt{2^{2}}}$

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

$x + \frac{3}{2} = \pm \frac{3 \sqrt{5}}{2}$

Subtract $\frac{3}{2}$ from both sides of the equation.

$x = \pm \frac{3 \sqrt{5}}{2} - \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{3 \sqrt{5}}{2} - \frac{3}{2}$

Decimal Form:

$x = 1.85410196 \dots, - 4.85410196 \dots$

Solve by Completing the Square x^2+3x-9=0

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