z4+10z2+9=0

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

u2+10u+9=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 9 and whose sum is 10.

1,9

Write the factored form using these integers.

(u+1)(u+9)=0

(u+1)(u+9)=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+1=0

u+9=0

Set the first factor equal to 0.

u+1=0

Subtract 1 from both sides of the equation.

u=-1

u=-1

Set the next factor equal to 0.

u+9=0

Subtract 9 from both sides of the equation.

u=-9

u=-9

The final solution is all the values that make (u+1)(u+9)=0 true.

u=-1,-9

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

z2=-1

(z2)1=-9

Solve the first equation for z.

z2=-1

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

z=±-1

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

Rewrite -1 as i.

z=±i

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=i

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

z=-i

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

z=i,-i

z=i,-i

z=i,-i

z=i,-i

Solve the second equation for z.

(z2)1=-9

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

(z2)1⋅11=-91

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

z2=-91

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

z=±-91

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

Simplify the right side of the equation.

Evaluate -91 as -9.

z=±-9

Rewrite -9 as -1(9).

z=±-1⋅9

Rewrite -1(9) as -1⋅9.

z=±-1⋅9

Rewrite -1 as i.

z=±i⋅9

Rewrite 9 as 32.

z=±i⋅32

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

z=±i⋅3

Move 3 to the left of i.

z=±3i

z=±3i

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=3i

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

z=-3i

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

z=3i,-3i

z=3i,-3i

z=3i,-3i

z=3i,-3i

The solution to z4+10z2+9=0 is z=i,-i,3i,-3i.

z=i,-i,3i,-3i

Solve for z z^4+10z^2+9=0