Problem 162 - Algebra and Trigonometry

Question

Describe the relationship between the real solutions of \(a x^{2}+b x+c=0\) and the graph of \(y=a x^{2}+b x+c\)

Step-by-Step Solution

Answer

The real solutions of the quadratic equation \(a x^{2}+b x+c=0\) are the x-coordinates where the graph of the function \(y=a x^{2}+b x+c\) intersects the x-axis.

Step 1: Understanding the Quadratic Equation and Function

Given a quadratic equation \(a x^{2}+b x+c=0\), to find its solutions, it is required to find the values of \(x\) for which the equation holds. Meanwhile, \(y=a x^{2}+b x+c\) is the function representation of the equation where the dependent variable, \(y\), varies with the independent variable, \(x\).

Step 2: Solving the Quadratic Equation

The solutions of the equation \(a x^{2}+b x+c=0\) are the roots of the equation, which are given by the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). These are the x-coordinates where the parabola intersects the x-axis.

Step 3: Graph of the Quadratic Function

The graph of a quadratic function \(y = a x^{2}+b x+c\) is a curve called a parabola. If \(a > 0\), the graph opens upwards, and if \(a < 0\), it opens downwards. The graph intersects the x-axis at the real solutions of the equation (roots of the function).

Step 4: Relationship between the Solutions and the Graph

The real solutions of the quadratic equation are the x-coordinates where the graph of the function intersects the x-axis (These are the roots). If the function has two real solutions, the graph will intersect the x-axis at two points. If the equation has one real solution, the graph will be tangent to the x-axis at that point. Lastly, if the quadratic equation has no real solutions, the graph will not touch or cross the x-axis at all.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This equation describes a parabola's behavior in algebra and provides insights into its geometry. To find the solutions, also known as the roots, we use:

\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]

This formula allows us to determine the x-values at which the parabolic curve intersects the x-axis. The quantity \( b^2 - 4ac \) within the square root is known as the discriminant, and it plays a crucial role in understanding the nature of the solutions.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real solution, meaning the parabola is tangent to the x-axis at one point.
• If negative, there are no real roots, and the parabola does not meet the x-axis.

Parabola Graph

A parabola is a U-shaped curve that is the graphic representation of a quadratic equation. The graph of \( y = ax^2 + bx + c \) is always a parabola, with its orientation depending on the sign of \( a \).

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

The highest or lowest point of the parabola is called the vertex. This vertex represents the maximum or minimum value of the function depending on the direction the parabola opens. Parabolas are symmetric, meaning they reflect evenly across a vertical line running through the vertex. Understanding this symmetry is crucial for graphing and solving quadratic equations.

Roots of a Quadratic Equation

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are the solutions where the equation is satisfied. These roots are found using the quadratic formula and are also where the parabola crosses or touches the x-axis.

• If the equation has two distinct real roots, the graph intersects the x-axis twice.
• If there is one repeated real root, the graph touches the x-axis at one point (tangent).
• If there are no real roots, the parabola never intersects the x-axis.

These intersections give valuable information about the solution's nature, which is visual and algebraic.

Discriminant

The discriminant is an expression found inside the quadratic formula's square root, represented as \( b^2 - 4ac \). It is crucial for understanding the nature of the roots of the quadratic equation.

• If \( b^2 - 4ac > 0 \), the equation has two real and distinct roots.
• If \( b^2 - 4ac = 0 \), it has exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the equation has no real roots, only complex ones.

The discriminant helps predict how the parabola will intersect with the x-axis or if it will at all. This insight can inform further observations or analyses about the function.

X-Axis Intersections

The x-axis intersections of a parabola described by \( y = ax^2 + bx + c \) indicate the actual values, or roots, where the quadratic equation equals zero. These intersections are visual markers of the roots on the graph.

• For two x-axis intersections, the graph has two real roots, and the discriminant is positive.
• If the parabola is tangent to the x-axis, there is one point of intersection, indicating one real root and a zero discriminant.
• Without any x-axis intersections, there are no real roots; thus, the discriminant is negative.

Understanding how the graph interacts with the x-axis facilitates grasping the full solution space of the quadratic equation.