z4-7z2-18=0

Substitute u=z2 into the equation. This will make the quadratic formula easy to use.

u2-7u-18=0

u=z2

Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -18 and whose sum is -7.

-9,2

Write the factored form using these integers.

(u-9)(u+2)=0

(u-9)(u+2)=0

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

u-9=0

u+2=0

Set the first factor equal to 0.

u-9=0

Add 9 to both sides of the equation.

u=9

u=9

Set the next factor equal to 0.

u+2=0

Subtract 2 from both sides of the equation.

u=-2

u=-2

The final solution is all the values that make (u-9)(u+2)=0 true.

u=9,-2

Substitute the real value of u=z2 back into the solved equation.

z2=9

(z2)1=-2

Solve the first equation for z.

z2=9

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

z=±9

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

Simplify the right side of the equation.

Rewrite 9 as 32.

z=±32

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

z=±3

z=±3

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

First, use the positive value of the ± to find the first solution.

z=3

Next, use the negative value of the ± to find the second solution.

z=-3

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

z=3,-3

z=3,-3

z=3,-3

z=3,-3

Solve the second equation for z.

(z2)1=-2

Take the 1th root of each side of the equation to set up the solution for z

(z2)1⋅11=-21

Remove the perfect root factor z2 under the radical to solve for z.

z2=-21

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

z=±-21

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

Simplify the right side of the equation.

Evaluate -21 as -2.

z=±-2

Rewrite -2 as -1(2).

z=±-1⋅2

Rewrite -1(2) as -1⋅2.

z=±-1⋅2

Rewrite -1 as i.

z=±i2

z=±i2

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

First, use the positive value of the ± to find the first solution.

z=i2

Next, use the negative value of the ± to find the second solution.

z=-i2

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

z=i2,-i2

z=i2,-i2

z=i2,-i2

z=i2,-i2

The solution to z4-7z2-18=0 is z=3,-3,i2,-i2.

z=3,-3,i2,-i2

Solve for z z^4-7z^2-18=0