91Ó°ÊÓ

When solving quadratic equations, the discriminant is a crucial concept. It helps determine the nature of the solutions to a quadratic equation. The discriminant is part of the quadratic formula, and it is symbolized by the letter \( D \), calculated as \( D = b^2 - 4ac \). In the context of the given equation \( 2x^2 + bx + 5 = 0 \), the discriminant becomes \( D = b^2 - 40 \).
Understanding the discriminant's value provides insight into the behavior of the solutions:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, also known as a repeated or double root.
• If \( D < 0 \), the equation does not have real solutions; instead, it has two complex conjugate solutions.

It is essential to determine the discriminant when solving quadratic equations, as it guides you whether or not real solutions exist for a given set of coefficients.

Real solutions of a quadratic equation are the values of \( x \) that make the equation true. For a quadratic function \( ax^2 + bx + c = 0 \) to have at least one real solution, the discriminant \( D \) must be nonnegative, i.e., \( D \geq 0 \).
In our exercise, where the quadratic equation is given as \( 2x^2 + bx + 5 = 0 \), we aim to find \( b \) values such that the equation has real solutions. Analyzing the discriminant \( b^2 - 40 \), we focus on the inequality \( b^2 \geq 40 \).

• Solving this inequality, we find that \( b^2 = 40 \) gives the boundary values.
• Taking the square root provides \( -\sqrt{40} \leq b \leq \sqrt{40} \).

Thus, the interval for \( b \) is defined by these square root values, ensuring at least one real solution exists wherever the discriminant is zero or positive.

Analyzing the coefficients in a quadratic equation \( ax^2 + bx + c = 0 \) allows one to formulate conjectures about the nature of its solutions. The coefficients \( a, b, \) and \( c \) play distinct roles:

• \( a \) is the leading coefficient; it determines the direction and steepness of the parabola.
• \( b \) is the linear coefficient; it affects the location of the axis of symmetry and vertex horizontal displacement.
• \( c \) is the constant term; it influences the vertical displacement (where the graph crosses the y-axis).

For the equation \( 2x^2 + bx + 5 = 0 \), \( a = 2 \) and \( c = 5 \) are fixed, leaving \( b \) to vary. This variability of \( b \) directly impacts the discriminant, \( D = b^2 - 40 \).
A key observation is that the value range for \( b \) dictates the possibility of real solutions. As concluded earlier, \( -\sqrt{40} \leq b \leq \sqrt{40} \) ensures at least one real solution. This insight into coefficient interaction provides groundwork for how small changes in \( b \) can significantly alter the solution set, thus underscoring the importance of understanding each coefficient's impact when analyzing quadratic equations.