Question4. $0=x^{4}-2 x^{2}+1$

Studdy Solution

STEP 1

1. The equation $0 = x^4 - 2x^2 + 1$ is a polynomial equation.
2. We need to solve for the values of $x$ that satisfy this equation.
3. The equation can be factored or solved using substitution methods.

STEP 2

1. Recognize the structure of the polynomial and consider substitution for simplification.
2. Solve the resulting quadratic equation.
3. Substitute back to find the solutions for $x$ .
4. Verify the solutions.

STEP 3

Recognize that the given polynomial $x^4 - 2x^2 + 1$ can be simplified by a substitution. Let $y = x^2$ .

STEP 4

Substitute $y = x^2$ into the equation to reduce it to a quadratic form:
$0 = y^2 - 2y + 1$

STEP 5

Solve the quadratic equation $y^2 - 2y + 1 = 0$ using factoring.
$y^2 - 2y + 1 = (y - 1)^2$
So, the equation becomes:
$(y - 1)^2 = 0$

STEP 6

Solve for $y$ by finding the roots of the factored equation:
$y - 1 = 0$
Therefore,
$y = 1$

STEP 7

Substitute back $y = 1$ into the original substitution $y = x^2$ :
$x^2 = 1$

STEP 8

Solve for $x$ by taking the square root of both sides:
$x = \pm 1$

STEP 9

Verify the solutions by substituting $x = 1$ and $x = -1$ back into the original equation $x^4 - 2x^2 + 1$ .
For $x = 1$ :
$1^4 - 2(1^2) + 1 = 1 - 2 + 1 = 0$
For $x = -1$ :
$(-1)^4 - 2(-1)^2 + 1 = 1 - 2 + 1 = 0$
Both solutions satisfy the original equation.
Solution: The solutions to the equation $0 = x^4 - 2x^2 + 1$ are $x = 1$ and $x = -1$ .

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