Problem 37 - Algebra 1: Concepts and Skills

Question

Consider for quadratic equation \(y=2 x^{2}+6 x-3\). What does the discriminant tell you about the graph of \(y=2 x^{2}+6 x-3 ?\)

Step-by-Step Solution

Answer

The discriminant of given quadratic equation is 60, which is greater than zero. Therefore, the graph of the equation \(y=2x^{2}+6x-3\) will intersect the x-axis in two distinct points, meaning the equation has two distinct real roots.

Step 1: Identify the Coefficients

Identify the coefficient values from the given quadratic equation. In the given equation \(y=2x^{2}+6x-3\), the coefficient 'a' is the value in front of \(x^{2}\), which is 2. The coefficient 'b' is the value in front of 'x', which is 6 and the coefficient 'c' is the constant term, which is -3.

Step 2: Calculate the Discriminant

Plug the values of a, b, and c into the formula of discriminant i.e, \(D = b^{2} - 4ac\). Now substitute a=2, b=6, c=-3 in the equation. This gives \(D = (6)^{2} - 4*(2)*(-3) = 36+24 = 60\).

Step 3: Interpret the Result

The discriminant is 60 which is greater than zero. This signifies that the given quadratic function consists of two distinct real roots. It indicates that the parabolic graph of this quadratic equation crosses the 'x' axis at two distinct points.

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are called quadratic because they involve terms raised to the second power. The standard form represents a parabola when graphed. Quadratic equations have many applications in real life, including physics, engineering, and finance.

When solving a quadratic equation, one of the primary tools is the **quadratic formula**:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The term under the square root, \( b^2 - 4ac \), is known as the discriminant, and it plays a crucial role in determining the characteristics of the solutions.

Graph of a Quadratic Function

A quadratic function is typically expressed as \( y = ax^2 + bx + c \). The graph of this function is a **parabola**.
Parabolas have unique properties:

• **Vertex:** The highest or lowest point of the parabola, depending on the direction in which it opens. For the equation \( y = ax^2 + bx + c \), the vertex can be calculated using the formula \( x = -\frac{b}{2a} \).
• **Direction:** The parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
• **Axis of Symmetry:** The line that passes through the vertex and divides the parabola into two mirror images, given by \( x = -\frac{b}{2a} \).

The roots of the quadratic equation correspond to the points where the parabola intersects the x-axis. Understanding how to plot and interpret these graphs is crucial for solving quadratic equations and analyzing real-world scenarios.

Real Roots

The **real roots** of a quadratic equation are the values of \( x \) that solve the equation \( ax^2 + bx + c = 0 \). These roots represent the x-values where the parabola intersects the x-axis on its graph.
To find these roots, we use the discriminant \( D = b^2 - 4ac \). The value of the discriminant tells us about the nature of the roots:

• If \( D > 0 \): The equation has two distinct real roots. The graph has two x-intercepts.
• If \( D = 0 \): There is exactly one real root, often called a double root, meaning the parabola touches the x-axis at one point.
• If \( D < 0 \): No real roots exist, as the parabola does not intersect the x-axis.

In our original exercise, we found the discriminant of the equation \( y = 2x^2 + 6x - 3 \) to be 60, which means there are two distinct real roots, confirming that the parabola crosses the x-axis at two separate points.