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Magazine Chapter 3: Problem 45
Solve the quadratic equation graphically. $$ 2.5 x^{2}=4.75 x-2.1 $$
Short Answer
Expert verified
Solutions are x = 0.6 and x = 1.4.
Step by step solution
01
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: \( 2.5x^2 - 4.75x + 2.1 = 0 \).
02
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 \( c = 2.1 \).
03
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 \( a = 2.5 > 0 \).
04
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.
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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鈥攃alled a parabola鈥攊ntersects the x-axis. These intersection points, known as x-intercepts, are the solutions to the equation.
The graphical approach is useful because:
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.
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:
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.
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:
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 \( c \).
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:
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.
