91Ó°ÊÓ

The discriminant is a crucial component when dealing with quadratic equations, particularly in solving polynomial inequalities. It is given by the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the standard form of a quadratic equation \(ax^2 + bx + c\). The discriminant can tell you:

• If it is positive: there are two distinct real roots.
• If it is zero: there is exactly one real root, or a double root.
• If it is negative: there are no real roots, only complex ones.

In this exercise, the discriminant is zero, which indicates a unique real solution (or a double root). This special case means that the quadratic touches the x-axis at just one point. This point is not only a root but is also significant for testing intervals in inequalities.

The quadratic formula is an essential tool for finding the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). It is derived from completing the square and is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to calculate the solutions of the equation, whether they are real or complex. The \(\pm\) sign indicates that there can be up to two roots - one by adding and one by subtracting the square root part. In our case, using \(a = 4\), \(b = -4\), and \(c = 1\), the formula yields just one root, \(x = 0.5\), since the discriminant is zero. This particular root is essential for plotting and solving the inequality, as it serves as the critical point.

Interval notation is a concise way of writing subsets of the real number line. For inequalities, like the ones involving quadratics, we use it to represent solution sets comprehensively. In this form, brackets and parentheses are used:

• Parentheses \(( )\) indicate that the endpoint is not included, representing "open intervals."
• Brackets \([ ]\) signify that the endpoint is included, representing "closed intervals."

For the inequality \(4x^2 - 4x + 1 \geq 0\), the solution is expressed in interval notation as \((-\infty, 0.5] \cup [0.5, \infty)\). Here, \(0.5\) is included due to the \(\geq\) sign, indicating equality holds, with the union \(\cup\) combining both intervals as part of the solution.

A critical point in the context of polynomial inequalities is a value that separates different behavior of the function. For quadratic functions, roots often serve as critical points. They are the values where the inequality can change from true to false, or vice versa.In our exercise, the critical point is \(x = 0.5\). This particular value is where the quadratic function \(4x^2 - 4x + 1\) equals zero. Testing intervals>

• Choose numbers on either side of the critical point to see if they satisfy the inequality.
• For \((-\infty, 0.5)\), try a number like \(x = 0\).
• For \((0.5, \infty)\), try \(x = 1\).

Both test points should satisfy the original inequality, affirming both intervals are part of the solution with a closed dot, indicating inclusion of \(x = 0.5\) in the solution set.