91Ó°ÊÓ

In the world of quadratic functions, the discriminant is a crucial tool. It helps us understand the nature of the roots of a quadratic equation. The formula for the discriminant, often denoted by \( D \), is \( b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are coefficients from a quadratic equation of the form \( ax^2 + bx + c = 0 \). The discriminant provides the following insights:

• If \( D > 0 \), the quadratic has two distinct real roots.
• If \( D = 0 \), there is one real root (or a repeated root).
• If \( D < 0 \), no real roots exist; instead, the roots are complex numbers.

In the exercise, ensuring the inequality \( at^2 + 5t + 4 > 0 \) for all real \( t \) means the expression cannot have real roots. This makes the discriminant negative, confirming no real number solutions can satisfy this quadratic inequality.

A quadratic function is a type of polynomial with a degree of 2. Its general form is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constant coefficients. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

In our scenario, we have the quadratic \( at^2 + 5t + 4 \). To maintain positivity over all real numbers, the parabola must never cross the x-axis, meaning it doesn't have real roots and therefore remains entirely above the x-axis. This visual understanding supports the concept of why the discriminator must be negative for the expression given in our exercise.

Solving inequalities involves determining the range of values that satisfy an inequality expression. Quadratic inequalities, like linear ones, can be solved through several steps but usually involve:

• Rewriting the inequality in the standard quadratic form.
• Finding the discriminant to determine the nature of the roots.
• Testing intervals around the roots (if they exist) to determine where the inequality holds true.

When dealing with inequalities that must be true for all real numbers, like \( at^2 + 5t + 4 > 0 \), ensuring that the function has no real roots is crucial. This can provide a clear boundary condition, such as in our exercise, where \( a > \frac{25}{16} \) ensures the inequality holds.

Real numbers are a fundamental part of mathematics, representing any value along the continuum on the number line. They include all the rational numbers, such as fractions and integers, and all the irrational numbers. Real numbers are used in virtually every field of math and many real-world applications.
A quadratic inequality, like the one in our exercise, must be considered over this broad domain. If the inequality holds for all real numbers, it means no real number can make the quadratic expression zero or negative. Thus, the challenge lies in setting conditions on the coefficients so that the quadratic always evaluates as positive, just like determining \( a > \frac{25}{16} \) in our particular problem.