# Solve for x x^4-11x^2+24=0

x4-11×2+24=0
Substitute u=x2 into the equation. This will make the quadratic formula easy to use.
u2-11u+24=0
u=x2
Factor u2-11u+24 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 24 and whose sum is -11.
-8,-3
Write the factored form using these integers.
(u-8)(u-3)=0
(u-8)(u-3)=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-8=0
u-3=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
u-8=0
Add 8 to both sides of the equation.
u=8
u=8
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
u-3=0
Add 3 to both sides of the equation.
u=3
u=3
The final solution is all the values that make (u-8)(u-3)=0 true.
u=8,3
Substitute the real value of u=x2 back into the solved equation.
x2=8
(x2)1=3
Solve the first equation for x.
x2=8
Solve the equation for x.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
x=±8
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite 8 as 22⋅2.
Factor 4 out of 8.
x=±4(2)
Rewrite 4 as 22.
x=±22⋅2
x=±22⋅2
Pull terms out from under the radical.
x=±22
x=±22
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.
x=22
Next, use the negative value of the ± to find the second solution.
x=-22
The complete solution is the result of both the positive and negative portions of the solution.
x=22,-22
x=22,-22
x=22,-22
x=22,-22
Solve the second equation for x.
(x2)1=3
Solve the equation for x.
Take the 1th root of each side of the equation to set up the solution for x
(x2)1⋅11=31
Remove the perfect root factor x2 under the radical to solve for x.
x2=31
Take the square root of both sides of the equation to eliminate the exponent on the left side.
x=±31
The complete solution is the result of both the positive and negative portions of the solution.
Evaluate 31 as 3.
x=±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.
x=3
Next, use the negative value of the ± to find the second solution.
x=-3
The complete solution is the result of both the positive and negative portions of the solution.
x=3,-3
x=3,-3
x=3,-3
x=3,-3
The solution to x4-11×2+24=0 is x=22,-22,3,-3.
x=22,-22,3,-3
The result can be shown in multiple forms.
Exact Form:
x=22,-22,3,-3
Decimal Form:
x=2.82842712…,-2.82842712…,1.73205080…,-1.73205080…
Solve for x x^4-11x^2+24=0

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