Problem 82 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 3 x^{2}=4-8 x $$

Step-by-Step Solution

Answer

The roots are between -3 and -2, and between 0 and 1.

Step 1: Rearrange the Equation

We need to bring the equation to its standard form. Start by moving all terms to one side of the equation: \[ 3x^2 + 8x - 4 = 0 \]

Step 2: Identify the Function to Graph

The equation \(3x^2 + 8x - 4 = 0\) can be represented as the function \(f(x) = 3x^2 + 8x - 4\). We will graph this quadratic function.

Step 3: Graph the Quadratic Function

Graph the function \(f(x) = 3x^2 + 8x - 4\) using graphing tools, ensuring to identify where the graph crosses the x-axis. These intersections are the roots of the equation.

Step 4: Determine the Intervals of the Roots

After graphing, if the exact roots are not integers, observe where the graph crosses the x-axis. Identify the consecutive integer intervals which the roots fall between. For this function, the roots appear to lie between \(-3\) and \(-2\), and between \(0\) and \(1\).

Key Concepts

Solving Equations by Graphing

When we're solving equations by graphing, our aim is to visualize the equation on a coordinate plane. Graphing a function helps us to find its roots, or solutions, where the graph intersects the x-axis. These points of intersection represent the values of \(x\) where the equation is true.

To solve by graphing:

• First, rearrange the equation into a standard form. For quadratic equations, the standard form is usually \(ax^2 + bx + c = 0\).
• Next, interpret the equation as a function. For example, if we have \(3x^2 + 8x - 4 = 0\), it can be seen as \(f(x) = 3x^2 + 8x - 4\).
• Use graphing tools or graph paper to plot the function. This often involves calculating and plotting several points to draw an accurate shape of the parabola.
• Look where the graph crosses the x-axis. These intersections are the solutions or 'roots' of the equation.

Graphing is a powerful visual tool, especially when the roots are not simple integers, making it easier to estimate non-exact solutions.

Quadratic Function

A quadratic function is one of the most commonly encountered polynomial functions in algebra. The general form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

Key characteristics of a quadratic function include:

• **Parabola Shape**: The graph of a quadratic function forms a curve called a parabola.
• **Vertex**: This is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.
• **Axis of Symmetry**: It is a vertical line that divides the parabola into two mirror-image halves.
• **Direction**: The parabola opens upwards if \(a > 0\) and downwards if \(a < 0\).

Quadratic functions are fundamental in modeling many physical phenomena and solve real-world problems effectively.

Roots of Equations

In mathematical terms, the roots of an equation are the values of the variable that make the equation true. For quadratic equations, roots can be either real or complex numbers, and they are sometimes called "solutions" or "zero points."

Understanding Roots

• The roots are found where the graph of the function crosses the x-axis.
• A quadratic equation can have:
  • Two distinct real roots
  • One real root, known as a double root
  • No real roots but two complex roots
• These roots can also be computed using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• When graphing, if the roots are not integers, we identify the approximate intervals between which the roots lie based on where the graph crosses the axis.

Understanding the roots provides insights into the behavior of the quadratic function and can be essential in many applications, from solving practical problems to understanding theoretical concepts.