Solve for z z^4-17z^2+16=0

$z^{4} - 17 z^{2} + 16 = 0$

Step 1

Substitute $u = z^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 17 u + 16 = 0$

$u = z^{2}$

Step 2

Factor $u^{2} - 17 u + 16$ using the AC method.

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 $16$ and whose sum is $- 17$ .

$- 16, - 1$

Write the factored form using these integers.

$(u - 16) (u - 1) = 0$

Step 3

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

$u - 16 = 0$

$u - 1 = 0$

Step 4

Set $u - 16$ equal to $0$ and solve for $u$ .

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

Add $16$ to both sides of the equation.

$u = 16$

Step 5

Set $u - 1$ equal to $0$ and solve for $u$ .

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Add $1$ to both sides of the equation.

$u = 1$

Step 6

The final solution is all the values that make $(u - 16) (u - 1) = 0$ true.

$u = 16, 1$

Step 7

Substitute the real value of $u = z^{2}$ back into the solved equation.

$z^{2} = 16$

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

Step 8

Solve the first equation for $z$ .

$z^{2} = 16$

Step 9

Solve the equation for $z$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{16}$

Simplify $\pm \sqrt{16}$ .

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

$z = \pm \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm 4$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$z = 4$

Next, use the negative value of the $\pm$ to find the second solution.

$z = - 4$

The complete solution is the result of both the positive and negative portions of the solution.

$z = 4, - 4$

Step 10

Solve the second equation for $z$ .

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

Step 11

Solve the equation for $z$ .

Remove parentheses.

$z^{2} = 1$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$z = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$z = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$z = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$z = 1, - 1$

Step 12

The solution to $z^{4} - 17 z^{2} + 16 = 0$ is $z = 4, - 4, 1, - 1$ .

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

Solve for z z^4-17z^2+16=0

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