Problem 11 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 2 x^{2}-2 x-3=0 $$

Step-by-Step Solution

Answer

Roots are between -1 and 0, and between 1 and 2.

Step 1: Understand the Equation

The given equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -2 \), and \( c = -3 \). Our task is to find the roots (solutions) of this equation by graphing it.

Step 2: Graph the Quadratic Function

First, express the quadratic equation as a function: \( y = 2x^2 - 2x - 3 \). Next, plot this function on a graph. It will be a parabola that opens upwards because the coefficient of \( x^2 \) is positive.

Step 3: Identify the x-intercepts

The x-intercepts of the graph correspond to the roots of the equation \( 2x^2 - 2x - 3 = 0 \). These are the points where the graph crosses the x-axis.

Step 4: Analyze the Graph for Exact Roots or Interval

Look for the x-intercepts on the graph. If they land exactly on integer values, those are your exact roots. If they land between integers, identify which consecutive integers the roots lie between.

Step 5: State the Roots

Upon graphing, the parabola intersects the x-axis between the points -0.5 and 1.5 and between the points 1.5 and 3.5. Thus, the roots are not exact integers. The consecutive integers between which the roots lie are \(-0.5, 0.5\) and \(1.5, 2.5\).

Key Concepts

Graphing Methods

Graphing methods are a visual way to solve quadratic equations, which involves plotting the equation as a graph on a coordinate plane. This can help you easily see where the solutions, or roots, of the equation are located. To do this, you'll first convert the quadratic equation into a function, typically written as \( y = ax^2 + bx + c \). Next, you plot this function on a graph paper or with a graphing tool. For example, consider the quadratic equation \( 2x^2 - 2x - 3 = 0 \). We express it in the form \( y = 2x^2 - 2x - 3 \) and graph it.

• The shape of the graph will be a parabola.
• Since the coefficient of \( x^2 \) is positive, the parabola opens upwards.

After plotting, the x-intercepts can be identified. These x-intercepts are crucial as they represent the solutions to the original quadratic equation. By using graphing methods, you're able to visually locate the approximate values of the roots.

Roots of Equations

The roots of a quadratic equation are the values for which the equation equals zero. They are also referred to as the solutions or x-intercepts of the graph representing the equation. For the equation \( 2x^2 - 2x - 3 = 0 \), finding the roots by graphing means identifying the x-values where the parabola intersects the x-axis.
These intersections signify the points where the equation achieves a value of zero.

• If the parabola crosses the x-axis at distinct points, those points are the roots.
• If it touches but does not cross, there's one root, counted twice (a repeated root).
• Lastly, if the parabola does not touch at all, there are no real roots.

For the given equation, the graph shows intersections between -0.5 and 0.5, and between 1.5 and 2.5 on the x-axis. This indicates the roots are not exact integers and lay in the intervals between these points.

Parabola

A parabola is the graph of a quadratic function, which can be represented by an equation in the form \( y = ax^2 + bx + c \). The parabola's shape is determined by the sign and value of the coefficient \( a \). When graphing, you'll notice specific characteristics of parabolas:

• If \( a > 0 \), the parabola opens upwards, similar to a smile.
• If \( a < 0 \), it opens downwards, much like a frown.

Furthermore, the vertex of the parabola is a key point that provides information about the graph's maximum or minimum.
In our example, \( y = 2x^2 - 2x - 3 \), the positive \( a \) value signals an upward-opening parabola. It curves downwards to its lowest point (the vertex), and then upwards again.
The parabola's intersection with the x-axis gives us the roots of the equation. Thus, understanding the properties of a parabola helps in interpreting and solving quadratic equations by graphing methods efficiently.