Precalculus Examples

$3 z^{2} + 6 z + 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $3$ out of $3 z^{2} + 6 z + 3$ .

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Step 1.1.1

Factor $3$ out of $3 z^{2}$ .

$3 (z^{2}) + 6 z + 3 = 0$

Step 1.1.2

Factor $3$ out of $6 z$ .

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

Step 1.1.3

Factor $3$ out of $3$ .

$3 (z^{2}) + 3 (2 z) + 3 (1) = 0$

Step 1.1.4

Factor $3$ out of $3 (z^{2}) + 3 (2 z)$ .

$3 (z^{2} + 2 z) + 3 (1) = 0$

Step 1.1.5

Factor $3$ out of $3 (z^{2} + 2 z) + 3 (1)$ .

$3 (z^{2} + 2 z + 1) = 0$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$3 (z^{2} + 2 z + 1^{2}) = 0$

Step 1.2.2

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

$2 z = 2 \cdot z \cdot 1$

Step 1.2.3

Rewrite the polynomial.

$3 (z^{2} + 2 \cdot z \cdot 1 + 1^{2}) = 0$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 1$ .

$3 {(z + 1)}^{2} = 0$

Step 2

Divide each term in $3 {(z + 1)}^{2} = 0$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 {(z + 1)}^{2} = 0$ by $3$ .

$\frac{3 {(z + 1)}^{2}}{3} = \frac{0}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 {(z + 1)}^{2}}{3} = \frac{0}{3}$

Step 2.2.1.2

Divide ${(z + 1)}^{2}$ by $1$ .

${(z + 1)}^{2} = \frac{0}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $3$ .

${(z + 1)}^{2} = 0$

Step 3

Set the $z + 1$ equal to $0$ .

$z + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$z = - 1$