Basic Math Examples

$4 n^{2} + 12 n + 9 = 0$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4 n^{2}$ as ${(2 n)}^{2}$ .

${(2 n)}^{2} + 12 n + 9 = 0$

Step 1.2

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

${(2 n)}^{2} + 12 n + 3^{2} = 0$

Step 1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 n = 2 \cdot (2 n) \cdot 3$

Step 1.4

Rewrite the polynomial.

${(2 n)}^{2} + 2 \cdot (2 n) \cdot 3 + 3^{2} = 0$

Step 1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 n$ and $b = 3$ .

${(2 n + 3)}^{2} = 0$

Step 2

Set the $2 n + 3$ equal to $0$ .

$2 n + 3 = 0$

Step 3

Solve for $n$ .

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Step 3.1

Subtract $3$ from both sides of the equation.

$2 n = - 3$

Step 3.2

Divide each term in $2 n = - 3$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 n = - 3$ by $2$ .

$\frac{2 n}{2} = \frac{- 3}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{- 3}{2}$

Step 3.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 3}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$n = - \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{3}{2}$

Decimal Form:

$n = - 1.5$

Mixed Number Form:

$n = - 1 \frac{1}{2}$