Solve by Factoring z^4-65z^2+64=0

$z^{4} - 65 z^{2} + 64 = 0$

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

${(z^{2})}^{2} - 65 z^{2} + 64 = 0$

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

$u^{2} - 65 u + 64 = 0$

Factor $u^{2} - 65 u + 64$ using the AC method.

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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 $64$ and whose sum is $- 65$ .

$- 64, - 1$

Write the factored form using these integers.

$(u - 64) (u - 1) = 0$

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

$(z^{2} - 64) (z^{2} - 1) = 0$

Rewrite $64$ as $8^{2}$ .

$(z^{2} - 8^{2}) (z^{2} - 1) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 8$ .

$(z + 8) (z - 8) (z^{2} - 1) = 0$

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

$(z + 8) (z - 8) (z^{2} - 1^{2}) = 0$

Factor.

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Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 1$ .

$(z + 8) (z - 8) ((z + 1) (z - 1)) = 0$

Remove unnecessary parentheses.

$(z + 8) (z - 8) (z + 1) (z - 1) = 0$

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

$z + 8 = 0$

$z - 8 = 0$

$z + 1 = 0$

$z - 1 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$z + 8 = 0$

Subtract $8$ from both sides of the equation.

$z = - 8$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$z - 8 = 0$

Add $8$ to both sides of the equation.

$z = 8$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$z + 1 = 0$

Subtract $1$ from both sides of the equation.

$z = - 1$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$z - 1 = 0$

Add $1$ to both sides of the equation.

$z = 1$

The final solution is all the values that make $(z + 8) (z - 8) (z + 1) (z - 1) = 0$ true.

$z = - 8, 8, - 1, 1$

Solve by Factoring z^4-65z^2+64=0

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