Precalculus Examples

${(z^{2} + 8 z)}^{2} + 23 (z^{2} + 8 z) + 112 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = z^{2} + 8 z$ . Substitute $u$ for all occurrences of $z^{2} + 8 z$ .

$u^{2} + 23 u + 112 = 0$

Step 1.2

Factor $u^{2} + 23 u + 112$ using the AC method.

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Step 1.2.1

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 $112$ and whose sum is $23$ .

$7, 16$

Step 1.2.2

Write the factored form using these integers.

$(u + 7) (u + 16) = 0$

Step 1.3

Replace all occurrences of $u$ with $z^{2} + 8 z$ .

$(z^{2} + 8 z + 7) (z^{2} + 8 z + 16) = 0$

Step 2

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

$z^{2} + 8 z + 7 = 0$

$z^{2} + 8 z + 16 = 0$

Step 3

Set $z^{2} + 8 z + 7$ equal to $0$ and solve for $z$ .

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Step 3.1

Set $z^{2} + 8 z + 7$ equal to $0$ .

$z^{2} + 8 z + 7 = 0$

Step 3.2

Solve $z^{2} + 8 z + 7 = 0$ for $z$ .

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Step 3.2.1

Factor $z^{2} + 8 z + 7$ using the AC method.

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Step 3.2.1.1

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 $7$ and whose sum is $8$ .

$1, 7$

Step 3.2.1.2

Write the factored form using these integers.

$(z + 1) (z + 7) = 0$

Step 3.2.2

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

$z + 1 = 0$

$z + 7 = 0$

Step 3.2.3

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

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Step 3.2.3.1

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

$z + 1 = 0$

Step 3.2.3.2

Subtract $1$ from both sides of the equation.

$z = - 1$

Step 3.2.4

Set $z + 7$ equal to $0$ and solve for $z$ .

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Step 3.2.4.1

Set $z + 7$ equal to $0$ .

$z + 7 = 0$

Step 3.2.4.2

Subtract $7$ from both sides of the equation.

$z = - 7$

Step 3.2.5

The final solution is all the values that make $(z + 1) (z + 7) = 0$ true.

$z = - 1, - 7$

Step 4

Set $z^{2} + 8 z + 16$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z^{2} + 8 z + 16$ equal to $0$ .

$z^{2} + 8 z + 16 = 0$

Step 4.2

Solve $z^{2} + 8 z + 16 = 0$ for $z$ .

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Step 4.2.1

Factor using the perfect square rule.

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Step 4.2.1.1

Rewrite $16$ as $4^{2}$ .

$z^{2} + 8 z + 4^{2} = 0$

Step 4.2.1.2

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

$8 z = 2 \cdot z \cdot 4$

Step 4.2.1.3

Rewrite the polynomial.

$z^{2} + 2 \cdot z \cdot 4 + 4^{2} = 0$

Step 4.2.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 4$ .

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

Step 4.2.2

Set the $z + 4$ equal to $0$ .

$z + 4 = 0$

Step 4.2.3

Subtract $4$ from both sides of the equation.

$z = - 4$

Step 5

The final solution is all the values that make $(z^{2} + 8 z + 7) (z^{2} + 8 z + 16) = 0$ true.

$z = - 1, - 7, - 4$