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

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
Factor u2-7u-18 using the AC method.
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 and solve.
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 and solve.
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
Solve the equation for z.
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
Solve the equation for z.
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

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