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Chapter 5: Problem 17

Use the related graph of each equation to determine its solutions. $$ -0.5 x^{2}=0 $$

Short Answer

Expert verified
The solution is \(x = 0\).

Step by step solution

01

Understand the Equation

We are given the quadratic equation \(-0.5x^2 = 0\). Our task is to find the solutions to this equation by using a graph. This means we need to find the value(s) of \(x\) that make this equation true.
02

Simplify the Equation

Since the equation is \(-0.5x^2 = 0\), we can divide both sides by \(-0.5\) to simplify it, resulting in \(x^2 = 0\). This means we need to find \(x\) such that \(x^2\) equals zero.
03

Graph the Equation

The equation \(x^2 = 0\) can be represented as the graph of the function \(y = x^2\). This is a parabola opening upwards that touches the x-axis at the origin, which is the point \((0,0)\).
04

Determine the Solutions From the Graph

The solution to the equation is the x-coordinate where the graph touches or crosses the x-axis. From the graph of \(y = x^2\), it only touches the x-axis at the origin, \((0,0)\). Thus, the solution of the equation \(x^2 = 0\) is \(x = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph of a Function
A graph of a function serves as a visual representation of an equation. It helps us understand how the values of the variables in a function relate to each other. In the context of a quadratic equation like \[-0.5x^2 = 0\] , the graph gives us insight into where the function equals zero.
  • This particular equation, when simplified, becomes \[x^2 = 0\] , which is represented by the graph of the function \[y = x^2\] .
  • The function \[y = x^2\] has a very distinctive shape known as a parabola. In our case, this parabola touches the x-axis at just one point, the origin. This tells us where the solutions of the equation lie.
  • Graphing functions can often make complex equations simpler to solve, as it provides a visual way to see solutions or check the behavior of the function.
To plot such graphs, one would typically plot several points of the function and connect them to see how the curve behaves. This aids in finding the precise solutions or roots of the equation.
Solutions of Equations
A solution of an equation refers to the value(s) that solve said equation, making it a true statement. Specifically, for a quadratic equation like \[-0.5x^2 = 0\] , solving the equation means finding the value(s) of \(x\) that satisfy the equation.
  • By simplifying the equation to \[x^2 = 0\] , we determine that \(x\) must be 0, since \(0^2 = 0\)
  • Solutions can often be found where the graph of an equation intersects the x-axis. The x-coordinates of these points are the solutions.
  • If a graph does not touch or cross the x-axis, then sometimes a quadratic equation has no real solutions, or it might have solutions in terms of complex numbers instead.
Understanding the concept of solutions is crucial because it allows us to find where equations hold true, and often, these solutions have practical applications in diverse areas such as physics, engineering, and finance.
Parabola
A parabola is an essential concept in algebra, especially when discussing quadratic equations. It is a U-shaped curve that can open upwards or downwards depending on the coefficient of the squared term, \(a\), in the standard quadratic form \[ax^2 + bx + c = 0\].
  • In the equation we analyzed, \[-0.5x^2 = 0\], once simplified to \[x^2 = 0\], the parabola \(y = x^2\) represents the graph which opens upwards, making the vertex its lowest point.
  • Parabolas are symmetric around a vertical line, known as the axis of symmetry, which for \[y = x^2\] is the y-axis itself.
  • Parabolas have a unique point called the vertex, which is the peak or the minimum of the parabola. In our case, the vertex is at the origin point \((0,0)\).
Recognizing a parabola, knowing how to graph it, and finding its vertex and axis of symmetry are integral skills in learning how to work with quadratic functions. This knowledge empowers students to solve equations by understanding the underlying geometry of the graph.

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