Algebra Examples

${(z^{2} + 4 z)}^{2} + 7 (z^{2} + 4 z) + 12 = 0$

Step 1

Let $u = z^{2} + 4 z$ . Substitute $u$ for all occurrences of $z^{2} + 4 z$ .

$u^{2} + 7 u + 12 = 0$

Step 2

Factor $u^{2} + 7 u + 12$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $7$ .

$3, 4$

Step 2.2

Write the factored form using these integers.

$(u + 3) (u + 4) = 0$

Step 3

Replace all occurrences of $u$ with $z^{2} + 4 z$ .

$(z^{2} + 4 z + 3) (z^{2} + 4 z + 4) = 0$

Step 4

Factor $z^{2} + 4 z + 3$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 4.2

Write the factored form using these integers.

$(z + 1) (z + 3) (z^{2} + 4 z + 4) = 0$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

$(z + 1) (z + 3) (z^{2} + 4 z + 2^{2}) = 0$

Step 5.2

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

$4 z = 2 \cdot z \cdot 2$

Step 5.3

Rewrite the polynomial.

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

Step 5.4

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

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

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z + 1 = 0$

$z + 3 = 0$

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

Step 7

Set $z + 1$ equal to $0$ and solve for $z$ .

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Step 7.1

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

$z + 1 = 0$

Step 7.2

Subtract $1$ from both sides of the equation.

$z = - 1$

Step 8

Set $z + 3$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $z + 3$ equal to $0$ .

$z + 3 = 0$

Step 8.2

Subtract $3$ from both sides of the equation.

$z = - 3$

Step 9

Set ${(z + 2)}^{2}$ equal to $0$ and solve for $z$ .

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Step 9.1

Set ${(z + 2)}^{2}$ equal to $0$ .

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

Step 9.2

Solve ${(z + 2)}^{2} = 0$ for $z$ .

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Step 9.2.1

Set the $z + 2$ equal to $0$ .

$z + 2 = 0$

Step 9.2.2

Subtract $2$ from both sides of the equation.

$z = - 2$

Step 10

The final solution is all the values that make $(z + 1) (z + 3) {(z + 2)}^{2} = 0$ true.

$z = - 1, - 3, - 2$