Solve by any method.

a^4 – 5a^2 + 4 = 0

I am having problems figuring out what method to use to solve this problem. I need to show my steps and I am not sure how to do this. Can someone help? Thanks.

This equation can be solved by factoring. Have you learned how to do that yet? You should have.

This is not college level math. It is taught in the ninth grade

Hint: one of the factors is (a-4)

OK, I made a mistake. Sorry about that. I didn't notice that is was fourth order.

The factors are:

(a^2-2)(a^2 +2)
which can be factored again to give
(a + sqrt2)(a - sqrt2)(a + isqrt2)(a - isqrt2)

Set that equal to zero and you will see that you have an answer whenever one of the factors is zero.
(i is the square root of -1)

To solve the equation a^4 - 5a^2 + 4 = 0, you can use factoring or substitution methods to find the values of a.

Option 1: Factoring
1. Rewrite the equation as follows: (a^2 - 4)(a^2 - 1) = 0.
2. Set each factor equal to zero:
a^2 - 4 = 0 or a^2 - 1 = 0.
3. Solve each factor separately:
For a^2 - 4 = 0, add 4 to both sides to get a^2 = 4. Then take the square root of both sides to find a = ±2.
For a^2 - 1 = 0, add 1 to both sides to get a^2 = 1. Then take the square root of both sides to find a = ±1.

Therefore, the solutions to the equation are a = -2, 2, -1, and 1.

Option 2: Substitution
1. Let y = a^2. Rewrite the equation as follows: y^2 - 5y + 4 = 0.
2. Solve the quadratic equation y^2 - 5y + 4 = 0 using factoring, completing the square, or the quadratic formula. In this case, factoring is the simplest method.
Factor the equation: (y - 1)(y - 4) = 0.
Set each factor equal to zero: y - 1 = 0 or y - 4 = 0.
3. Solve each equation separately:
For y - 1 = 0, add 1 to both sides to get y = 1.
For y - 4 = 0, add 4 to both sides to get y = 4.
4. Substitute back y = a^2 and solve for a:
For y = 1, substitute a^2 = 1 and take the square root of both sides to find a = ±1.
For y = 4, substitute a^2 = 4 and take the square root of both sides to find a = ±2.

Hence, the solutions to the equation are a = -2, 2, -1, and 1.