91Ó°ÊÓ

When dealing with quadratic equations, one important aspect is the discriminant. This term is found within the quadratic formula and plays a critical role in determining the type and number of solutions. The quadratic equation is generally written as \(ax^2 + bx + c = 0\). The discriminant is calculated using the formula \(b^2 - 4ac\).

Depending on the value of the discriminant, we can predict the nature of the solutions:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), the equation has exactly one real solution, often referred to as a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation has no real solutions but two complex solutions.

In our exercise, substituting the values of \(a\) and \(c\) in the formula helped find the intervals for which \(b\) results in real solutions.

Real solutions in the context of quadratic equations refer to the values of \(x\) that satisfy the equation, producing a real number. Simply put, these solutions lie on the real number line and do not involve imaginary numbers.

This means when we compute the solutions using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the expression under the square root—known as the discriminant—must be non-negative for real solutions to exist.

Real solutions give insight into various real-world problems and scenarios modeled by quadratic equations, providing tangible results.

• When the discriminant is zero, the real solution, also called a repeated root, indicates the quadratic graph touches the x-axis at one point.
• When two distinct real solutions are present, the graph crosses the x-axis at two points.

In the original problem, we identify what values of \(b\) ensure the equation has at least one real solution by setting the discriminant \(b^2 - 4\times3\times10 \geq 0\).

In a quadratic equation \(ax^2 + bx + c = 0\), the coefficients \(a\), \(b\), and \(c\) play a significant role in determining the nature of its solutions, along with the shape and position of its graph on a coordinate plane. Let's break down what each coefficient does:

• Coefficient \(a\): It determines the direction of the parabola. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• Coefficient \(b\): This affects the position of the parabola along the x-axis. Changing \(b\) shifts the roots of the parabola along the axis without altering its basic shape.
• Coefficient \(c\): It represents the y-intercept of the graph. The term \(c\) tells us where the parabola will cross the y-axis.

In our specific problem, the task was to find intervals for \(b\) that allow for real solutions. Using our understanding from the discriminant and how coefficients affect a quadratic equation, we learnt that the relationship \(b^2 \geq 4ac\) is key to achieving real solutions.