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Magazine Chapter 5: Problem 20
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}-3 x=0 $$
Short Answer
Expert verified
The roots of the equation are \(x = 0\) and \(x = 3\).
Step by step solution
01
Understand the Equation
The given equation is a quadratic equation: \(x^2 - 3x = 0\). Our goal is to find the roots of this quadratic equation by graphing.
02
Rearrange the Equation to Standard Form
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). The given equation \(x^2 - 3x = 0\) is already in this form with \(a=1\), \(b=-3\), and \(c=0\).
03
Determine the Parabola's Vertex and Axis of Symmetry
The vertex form of a parabola is given by \(x = -\frac{b}{2a}\). For this equation, \(-\frac{-3}{2 \cdot 1} = \frac{3}{2}\). This means the axis of symmetry is \(x = \frac{3}{2}\). The vertex can be found by substituting this into the equation, which confirms \(x = 0\) at \(y = 0\).
04
Graph the Equation
To graph \(y = x^2 - 3x\), plot the vertex and a few more points by choosing values for \(x\) and calculating \(y\). For example, when \(x = 0\), \(y = 0\), and when \(x = 3\), \(y=0\). This tells us the roots are at \(x = 0\) and \(x = 3\).
05
Identify the Roots from the Graph
From the graph, observe where the parabola crosses the x-axis. These points are the roots of the equation. The graph indicates that the parabola crosses the x-axis at \(x=0\) and \(x=3\).
06
Verify the Roots
Plug the roots back into the original equation to verify. For \(x = 0\), the equation becomes \(0^2 - 3 \times 0 = 0\), which is true. For \(x = 3\), the equation becomes \(3^2 - 3 \times 3 = 0\), which is also true. Thus, the roots are verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Method
The graphing method is a visual technique used to solve quadratic equations by plotting the equation on a coordinate plane. The main goal is to find where the curve, typically a parabola for quadratic equations, intersects the x-axis. These intersection points are known as "roots". To do this, let's break it down into simple steps:
Graphing makes it easier to visually understand solutions, especially helpful when the roots are not exact numbers.
- Identify the Equation: Begin with the given quadratic equation, for our exercise, it's \(x^2 - 3x = 0\).
- Convert to Standard Form: Ensure the equation is in the form \(ax^2 + bx + c = 0\). Our equation is already in this form with \(a = 1\), \(b = -3\), and \(c = 0\).
- Plot the Graph: Calculate sufficient points by selecting values for \(x\) and determining \(y\) using the equation \(y = x^2 - 3x\). Plot these points on a graph.
- Locate the Roots: Observe the graph to see where the parabola crosses the x-axis, which represents the roots of the equation.
Graphing makes it easier to visually understand solutions, especially helpful when the roots are not exact numbers.
Parabola
A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic equation. It's crucial for understanding quadratic equations graphically.When dealing with a quadratic equation like \(x^2 - 3x = 0\), the graph of the equation will produce a parabola due to the squared \(x\) term.There are key characteristics of parabolas you should know:
Recognizing these features makes analyzing the graph more intuitive, enabling us to easily identify where it crosses the x-axis, giving us the equation's roots.
- Vertex: This is the highest or lowest point of the parabola. For the given equation, the vertex is found using \(-\frac{b}{2a}\). So, \(-\frac{-3}{2 \times 1} = \frac{3}{2}\).
- Axis of Symmetry: It's a vertical line that divides the parabola into two mirror-image halves. In this case, it occurs at \(x = \frac{3}{2}\).
- Direction: The parabola opens upward when the coefficient of \(x^2\) is positive, as in our exercise, and downward when it's negative.
Recognizing these features makes analyzing the graph more intuitive, enabling us to easily identify where it crosses the x-axis, giving us the equation's roots.
Roots of Equations
The roots of a quadratic equation are the solutions or "zeroes" where the graph touches or intersects the x-axis. They are crucial because they represent the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).In the exercise example, \(x^2 - 3x = 0\), the roots are \(x = 0\) and \(x = 3\). Let's delve into why this matters:
Understanding the roots allows you to solve quadratic equations effectively and is fundamental in various applications like physics, engineering, and economics.
- Finding Roots: Utilize methods like factoring, completing the square, using the quadratic formula, or graphing (as done here) to find the roots.
- Verification: Always check roots by substituting back into the original equation. For \(x = 0\), we see \(0^2 - 3(0) = 0\) is true, and for \(x = 3\), \(3^2 - 3(3) = 0\) holds true.
- Interpreting Roots: Real roots imply exact intersection points, while non-real (complex) roots arise when the parabola doesn't intersect the x-axis.
Understanding the roots allows you to solve quadratic equations effectively and is fundamental in various applications like physics, engineering, and economics.
