91Ó°ÊÓ

When graphing inequalities, our main goal is to identify where the equation is positive, negative, or zero. Doing this allows us to determine the intervals where the inequality holds true. For the given inequality, we begin by plotting the function \( f(x) = x^4 - 2x^2 - 1 \) on a coordinate plane. The graph visually demonstrates which parts of the x-axis the inequality holds.

Areas where the graph is above the x-axis indicate solutions to \( x^4 - 2x^2 - 1 > 0 \). It's essential to precisely examine regions close to the x-axis to find transition points—the points where the graph changes from negative to positive (or vice versa). These are often found between integers and indicate the approximate roots we need. Software or a graphing calculator can aid in zooming into these transitions, making it easier to estimate the endpoints to the nearest thousandth.

• Plot the expression on a graph.
• Observe where the graph is above the x-axis.
• Estimate endpoints by checking any visible transitions between positive and negative values.

Quadratic transformation is a useful technique when dealing with polynomial inequalities that have powers of four. In this case, we aim to transform the polynomial equation into a more straightforward quadratic form. Using the substitution \( y = x^2 \), the given polynomial \( x^4 - 2x^2 - 1 \) can be rewritten as a quadratic equation: \( y^2 - 2y - 1 = 0 \).

This transformation simplifies the process of finding roots by reducing a complex degree-four equation to a more manageable degree-two equation. The quadratic formula \( y = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) is then employed to determine the roots for \( y \). From these roots, we revert back to our original variable \( x \), leading us to the possible solutions for the inequality.

• Use substitution \( y = x^2 \) to simplify the equation.
• Transform the original polynomial into a quadratic form.
• Solve this quadratic using the quadratic formula for clear solutions.

Solving inequalities algebraically involves finding exact solutions or roots algebraically. We do this by setting the transformed quadratic \( y^2 - 2y - 1 = 0 \) equal to zero. Solving it using the quadratic formula offers direct calculation of precise root values. Let's solve:
We first identify the coefficients: \( a = 1 \), \( b = -2 \), and \( c = -1 \).
With the formula \[ y = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \],
our calculation yields \[ y = \frac{2 \pm \sqrt{8}}{2} \], which simplifies down to \[ y = 1 \pm \sqrt{2} \].
From these results:

• \( x^2 = 1 + \sqrt{2} \) leads to \( x = \pm\sqrt{1+\sqrt{2}} \).
• \( x^2 = 1 - \sqrt{2} \) doesn't yield real solutions, as its value is negative.

Thus, verified real roots are \( x = \pm\sqrt{1+\sqrt{2}} \). Calculators help confirm that these are accurate solutions, corresponding to root approximations obtained graphically.