Find all real numbers x which satisfy the equation. 10x6−10x2=0

Answer 1

Factor

10x⁶ - 10x² = 10x²(x⁴ - 1) = 10x²(x² - 1)(x² + 1) = 10x²(x - 1)(x + 1)(x² + 1)

Then 10x⁶ - 10x² = 0 if and only if 10x²(x - 1)(x + 1)(x² + 1) = 0. Since x² + 1 has no real roots, 10x²(x - 1)(x + 1) = 0. Therefore x = 0, 1, or -1.

Answer 2

I need to solve the equation 10x⁶ - 10x² = 0.

Let me factor this equation step by step.

Step 1: Factor out the common terms

10x⁶ - 10x² = 0

First, I can factor out 10x²:

10x²(x⁴ - 1) = 0

Step 2: Factor x⁴ - 1

I can recognize that x⁴ - 1 is a difference of squares:

x⁴ - 1 = (x²)² - 1² = (x² - 1)(x² + 1)

And x² - 1 is also a difference of squares:

x² - 1 = (x - 1)(x + 1)

So the equation becomes:

10x²(x - 1)(x + 1)(x² + 1) = 0

Step 3: Solve for x

For this product to equal zero, at least one factor must be zero:

1. 10x² = 0 → x² = 0 → x = 0
2. x - 1 = 0 → x = 1
3. x + 1 = 0 → x = -1
4. x² + 1 = 0 → x² = -1 → This has no real solutions (only complex solutions x = ±i)

Answer:

The real solutions are: x = -1, 0, 1

Or in set notation: {-1, 0, 1}

Answer 3

10x^6 - 10x^2 = 0

divide by 10

x^6 -x^2 = 0

factor out x^2

x^2(x^4-1) = 0

x^2(x^2-1)(x^2+1) = 0

x^2(x+1)(x-1)(x^2+1)

roots=solutions= 0, -1, +1, and i & -i