Can somebody please help me with these problems?
Factor the expression on the left side of each equation. Then solve.
1)x^4-5x^2+4=0
2)x^4-8x^2+16=0
Please and thank you!
1) let y = x^2, then
y^2-5y+4=0
(y-4)(y-1)=0
y = 4 or y = 1
x^2 = 4 or x^2 = 1
x = -2, +2, -1, +1
same approach for #2
So the factored form would be (y-4) (y-1) and the solutions would be -2, 2, -2, and 1?
We learned that you were supposed to do it like this:
x^4-5x^2+4=0
(x^2)^2-5(x^2)+4
a^2-2a+4...and then factor it and I'm not quite sure how to do that. Does it make any sense to you?
Okay, I'm Valerie by the way. Actually, I'm Linda and Valerie. But actually, I'm neither. I just always change my name. Sorry about that.
No problem, Linda and Valerie! I can definitely help you with those problems.
For the first problem:
We have the equation x^4 - 5x^2 + 4 = 0, and you're right, we can use a substitution to make it easier. Let's substitute y = x^2. Then the equation becomes y^2 - 5y + 4 = 0.
To factor this quadratic equation, we want to find two numbers that multiply to give 4, and add up to give -5. Those numbers are -4 and -1:
(y - 4)(y - 1) = 0.
Now, we have two solutions: y - 4 = 0, which gives y = 4, and y - 1 = 0, which gives y = 1. Remember, y is just a substitution for x^2.
Substituting y back in with x^2, we have:
x^2 = 4 or x^2 = 1.
Taking the square root of both sides:
x = ±2 or x = ±1.
So, the solutions to the equation x^4 - 5x^2 + 4 = 0 are x = -2, 2, -1, 1.
Great work on that! Let's move on to the second problem.
We have x^4 - 8x^2 + 16 = 0.
Following the same approach as before, let's substitute y = x^2:
y^2 - 8y + 16 = 0.
This equation factors nicely as a perfect square trinomial: (y - 4)(y - 4) = 0.
Therefore, we have only one solution: y - 4 = 0, which gives y = 4.
Substituting y back in with x^2, we have:
x^2 = 4.
Taking the square root of both sides:
x = ±2.
So, the solution to the equation x^4 - 8x^2 + 16 = 0 is x = -2, 2.
I hope this helps you understand how to factor and solve these problems. Feel free to ask any further questions!