Factorise x⁴ - 5x² + 4 < 0

To factor the expression \(x^4 - 5x^2 + 4 < 0\), we can start by making a substitution. Let \(y = x^2\). This transforms the inequality into:

Rewriting the Inequality

The expression becomes:

y^2 - 5y + 4 < 0

Factoring the Quadratic

Next, we need to factor the quadratic equation:

y^2 - 5y + 4 = (y - 1)(y - 4)

Setting Up the Inequality

Now, we rewrite the inequality:

(y - 1)(y - 4) < 0

Finding the Critical Points

The critical points occur when each factor equals zero:

• y - 1 = 0 → y = 1
• y - 4 = 0 → y = 4

Analyzing the Intervals

We will test the intervals determined by these critical points: (-∞, 1), (1, 4), and (4, ∞).

• For y < 1 (e.g., y = 0): (0 - 1)(0 - 4) = 4 > 0
• For 1 < y < 4 (e.g., y = 2): (2 - 1)(2 - 4) = -2 < 0
• For y > 4 (e.g., y = 5): (5 - 1)(5 - 4) = 4 > 0

Conclusion of the Analysis

The inequality (y - 1)(y - 4) < 0 holds true for the interval:

1 < y < 4

Returning to x

Substituting back \(y = x^2\), we have:

1 < x^2 < 4

Final Result

This means:

• -2 < x < -1
• 1 < x < 2

Thus, the solution to the inequality \(x^4 - 5x^2 + 4 < 0\) is:

x ∈ (-2, -1) ∪ (1, 2)