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

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.

u-8=0

Add 8 to both sides of the equation.

u=8

u=8

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

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

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