X^4-5X^2+4=0
Substitute u=X^2 into the equation. This will make the quadratic formula easy to use.
u^2-5u+4=0
u=X^2
Move -4 to the left-hand side of the equation by adding it to both sides. The goal is to have all terms on the left-hand side equal to 0.
u^2-5u+4=0
In this problem -1*-4=4 and -1-4=-5, so insert -1 as the right hand term of one factor and -4 as the right-hand term of the other factor.
(u-1)(u-4)=0
Set each of the factors of the left-hand side of the equation equal to 0.
u-1=0
u-4=0
Since -1 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 1 to both sides.
u=1
u-4=0
Set each of the factors of the left-hand side of the equation equal to 0.
u=1
u-4=0
Since -4 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 4 to both sides.
u=1
u=4
The complete solution is the set of the individual solutions.
u=1, 4
Substitute the real value of u=X^2 back into the solved equation.
X^2=1
X^2=4
Solve the first equation for X.
X^2=1
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
X=+/-sqrt(1)
Pull all perfect square roots out from under the radical. In this case, remove the 1 because it is a perfect square.
X=+/-1
First, substitute in the + portion of the +/- to find the first solution.
X=1
Next, substitute in the - portion of the +/- to find the second solution.
X=-1
The complete solution is the result of both the + and - portions of the solution.
X=1,-1
Solve the second equation for X.
X^2=4
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
X=+/-sqrt(4)
Pull all perfect squXre roots out from under the radical. In this case, remove the 2 because it is a perfect square.
X=+/-2
First, substitute in the + portion of the +/- to find the first solution.
X=2
Next, substitute in the - portion of the +/- to find the second solution.
X=-2
The complete solution is the result of both the + and - portions of the solution.
X=2,-2
The solution to X^4-5X^2+4=0 is X=1,-1,2,-2.
X=1,-1,2,-2