Problem 45 - College Algebra with Modeling and Visualization

Question

Solve the quadratic equation graphically. $$ 2.5 x^{2}=4.75 x-2.1 $$

Step-by-Step Solution

Answer

Solutions are x = 0.6 and x = 1.4.

Step 1: Rewrite the Equation in Standard Form

A quadratic equation is typically written in the standard form as \( ax^2 + bx + c = 0 \). Let's start by rearranging the given equation \( 2.5x^2 = 4.75x - 2.1 \). Subtract \( 4.75x \) and add \( 2.1 \) from both sides to get: \( 2.5x^2 - 4.75x + 2.1 = 0 \).

Step 2: Identify the Coefficients

From the standard form of the quadratic equation \( 2.5x^2 - 4.75x + 2.1 = 0 \), identify the coefficients: \( a = 2.5 \), \( b = -4.75 \), and \( c = 2.1 \).

Step 3: Graph the Quadratic Function

To solve the equation graphically, consider the function \( y = 2.5x^2 - 4.75x + 2.1 \). Plot this quadratic curve on a graphing calculator or graphing software. The parabola will be upward-opening because \( a = 2.5 > 0 \).

Step 4: Find the x-Intercepts

The x-intercepts of the graph represent the solutions to the equation \( 2.5x^2 - 4.75x + 2.1 = 0 \). Look for points where the curve crosses the x-axis. Let's say the graphical solution shows intersections at two points.

Step 5: Interpret Graphical Solutions

The x-coordinates of the points where the graph intersects the x-axis are the solutions to the quadratic equation. Ensure the intercepts are accurately read from the graph. For example, if it intersects at \( x = 0.6 \) and \( x = 1.4 \), these are the solutions.

Key Concepts

Graphical Solutions

Graphical solutions provide a visual way to find the roots of a quadratic equation. By plotting the equation on a graph, you can see where the curve—called a parabola—intersects the x-axis. These intersection points, known as x-intercepts, are the solutions to the equation.

The graphical approach is useful because:

• It offers a clear visual representation of the solutions by showing where the curve meets the x-axis.
• This method helps in understanding the behavior of the curve, like how it opens and where it is positioned relative to the axes.

While graphical solutions are intuitive, they require accurate plotting. Often, this method is supported by graphing tools or software to ensure precision in reading the intercepts.

Coefficient Identification

Coefficient identification is a critical step in solving quadratic equations, especially when graphing. Quadratic equations are generally in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients that define the shape and position of the parabola.

Here's how identifying coefficients helps:

• \( a \): Determines the direction of the parabola. If \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.
• \( b \): Influences the symmetry and position; affects how the parabola shifts horizontally.
• \( c \): Represents the y-intercept of the quadratic function; influences the vertical position of the parabola.

By clearly identifying these coefficients, you can accurately graph the parabola and understand its properties.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). It is essential for simplifying and solving quadratic equations, both algebraically and graphically.

Converting an equation into its standard form involves:

• Rearranging terms to ensure \( x^2 \), \( x \), and constant terms are clearly isolated.
• Makes it easier to apply formulas and graph, by clearly identifying \( a \), \( b \), and \( c \).

For the given equation \( 2.5x^2 = 4.75x - 2.1 \), you rearrange it into \( 2.5x^2 - 4.75x + 2.1 = 0 \). This set-up facilitates identifying the parabola's shape and direction when graphing.

Graphing Quadratics

Graphing quadratics involves plotting the equation \( y = ax^2 + bx + c \) on a Cartesian coordinate system to visualize its solution. The graph takes the form of a parabola, which showcases unique features based on the values of \( a \), \( b \), and \( c \).

Key elements in graphing quadratics include:

• Direction: The value of \( a \) indicates whether the parabola opens upward or downward.
• Vertex: The point where it changes direction, found using \( x = -\frac{b}{2a} \).
• Intercepts: Points where it crosses the x-axis (solutions) and y-axis.

When you understand these elements, plotting and interpreting the graph becomes simpler, allowing you to find solutions to the quadratic equation efficiently.