Problem 10 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 4 x^{2}-7 x-15=0 $$

Step-by-Step Solution

Answer

Roots are between \(-1\text{ and }0\) and \(3\text{ and }4\).

Step 1: Identify the function

The equation given is \( 4x^2 - 7x - 15 = 0 \). This is a quadratic equation. To solve it by graphing, we need to consider it as a function: \( y = 4x^2 - 7x - 15 \).

Step 2: Create a table of values

Choose a range of values for \( x \), both negative and positive, and calculate corresponding \( y \) values using \( y = 4x^2 - 7x - 15 \). For example, if \( x = -3, -2, -1, 0, 1, 2, 3 \), compute each \( y \).

Step 3: Plot the points on a graph

Using the calculated \( (x, y) \) points, plot them on a coordinate plane. This will form the parabola of the function \( y = 4x^2 - 7x - 15 \).

Step 4: Identify the x-intercepts

The x-intercepts are the points where the curve crosses the x-axis, i.e., where \( y = 0 \). These are the solutions to the equation \( 4x^2 - 7x - 15 = 0 \).

Step 5: Estimate the roots

From the graph, observe the x-coordinates where the parabola crosses the x-axis. These will not provide exact values but will give the approximate intervals. For instance, the graph suggests roots between \( x = -1 ext{ and } 0 \) and \( x = 3 ext{ and } 4 \).

Step 6: State the consecutive integers

Since the exact roots could not be easily found, state the integers between which the roots are located based on the x-intercepts: between \(-1 ext{ and }0\) and \(3 ext{ and }4\).

Key Concepts

Graphing Quadratic Functions

When solving quadratic equations, graphing the related quadratic function is a useful visual method. A quadratic function typically appears as a parabola on a coordinate plane, due to its characteristic equation form, \( ax^2 + bx + c = 0 \). To graph a quadratic function, follow these steps:

• Identify the quadratic equation, such as \( y = 4x^2 - 7x - 15 \).
• Create a table of values by selecting a range of \( x \) values to compute the corresponding \( y \) values.
• Plot these points on a graph. The shape that emerges is a parabola.

Ensure to include both negative and positive \( x \) values to get a comprehensive view of the parabola's shape.

X-Intercepts

Finding the x-intercepts of a quadratic function graph is crucial for solving the equation. The x-intercepts are points on the graph where \( y = 0 \). In essence, these points represent the solutions or roots of the quadratic equation.

• After plotting the parabola, look for the points where it crosses the x-axis.
• These x-values are your estimated solutions to \( 4x^2 - 7x - 15 = 0 \).

Precise x-intercepts can be tricky to determine through graphing alone, often providing an interval rather than exact values.

Parabolas

Parabolas are U-shaped curves that graphically represent quadratic functions. The main features of a parabola include:

• Vertex: The lowest or highest point, depending on the opening direction.
• Axis of symmetry: A vertical line that divides the parabola into two mirror-image halves.
• Direction of opening: Determined by the sign of the coefficient \( a \). If \( a > 0 \), it opens upwards; if \( a < 0 \), downwards.

Understanding the components of a parabola helps in predicting its shape and position on the graph.

Consecutive Integers

In some cases, the x-intercepts of a parabola as observed on a graph are not clearly defined. This often occurs when the roots are not whole numbers. Here, stating consecutive integers provides an approximate solution.

• If the parabola crosses between \( x = -1 \) and \( x = 0 \), say the root is between these integers.
• The same method applies if it crosses between \( x = 3 \) and \( x = 4 \).

Specifying consecutive integers helps to give a clearer idea of where the exact roots might be within an interval.