Problem 36 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The graph intersects at \( x = -4 \) and between \( x = 0 \) and \( x = 1 \).

Step 1: Understand the Equation

The given equation is a quadratic equation, \( x^2 + 3x - 2 = 0 \). Our goal is to find the roots of this equation by graphing the function \( f(x) = x^2 + 3x - 2 \).

Step 2: Write the Function Form

Express the quadratic equation in the form of a function: \( f(x) = x^2 + 3x - 2 \). This will allow us to graph it and find where it intersects the x-axis to determine the roots.

Step 3: Sketch the Graph of the Quadratic Function

To graph the function \( f(x) = x^2 + 3x - 2 \), we need to plot several points on a coordinate plane. Choose values of \( x \), calculate corresponding \( f(x) \) values, and plot these points. Then, draw a smooth curve through these points, creating a parabola that opens upwards.

Step 4: Identify the Intersection with the x-axis

Examine where the graphed parabola intersects the x-axis. These points are the roots of the equation \( x^2 + 3x - 2 = 0 \). The intersections are where \( f(x) = 0 \).

Step 5: Determine the Exact Roots or Approximate

If the intersections with the x-axis are clear points, they are the exact roots. If not precise, use the graph to see between which consecutive integers each intersection lies.

Step 6: Finalize the Roots

Upon examining the graph, the parabola crosses the x-axis at approximately \( x = -4 \) and \( x = 0.5 \). Therefore, the exact roots or the intervals between which they lie: the roots are approximately \( x = -4 \) and a value between \( x = 0 \) and \( x = 1 \).

Key Concepts

Graphing Functions

Graphing a quadratic function is a powerful visual tool for understanding how the function behaves. Start by identifying the standard form of the quadratic equation, which is typically given as \( ax^2 + bx + c = 0 \). In our case, the function \( f(x) = x^2 + 3x - 2 \) represents a quadratic equation. To graph this function, select several values for \( x \) across a range that includes both negative and positive values.
Calculate the corresponding \( f(x) \) values to create coordinate pairs. Plot these points neatly on a graph paper or using graphing software.
By connecting these plotted points with a smooth, continuous curve, a parabola forms, revealing the nature and direction of the quadratic function.

• The vertex of the parabola provides insight into the minimum or maximum point of the function.
• The direction (upward or downward opening) is influenced by the coefficient of \( x^2 \).

When the parabola intersects the x-axis, it helps us determine the roots of the equation.

Roots of Equations

The roots of a quadratic equation are essential to understanding its solutions. Roots refer to the values of \( x \) for which \( f(x) = 0 \). For our quadratic function \( f(x) = x^2 + 3x - 2 \), these roots represent the points where the graph intersects the x-axis.
There are different types of roots you can encounter:

• **Real and Distinct Roots:** Two separate points where the parabola crosses the x-axis.
• **Real and Repeated Roots:** A single point of intersection, also known as a double root or vertex.
• **Complex Roots:** No points of intersection with the x-axis, indicating the parabola does not touch or intersect the x-axis at all.

In this case, the graph shows the parabola intersecting at two points: \( x = -4 \) and a point between \( x = 0 \) and \( x = 1 \). Identifying these roots helps solve the quadratic equation.

Parabolas

Parabolas are curved, symmetrical shapes that graphically represent quadratic functions. The specific form of a parabola for the equation \( f(x) = x^2 + 3x - 2 \) results from the nature of quadratic equations. Parabolas have distinct characteristics that make them easy to recognize:

• **Vertex:** The tip of the parabola, which acts as its point of symmetry. It's crucial for identifying the maximum or minimum value of a function.
• **Axis of Symmetry:** A vertical line running through the vertex of the parabola, dividing it into two mirror-image halves.
• **Direction:** Determined by the leading coefficient (the one before \( x^2 \)). If positive, the parabola opens upwards; if negative, it opens downwards.

The understanding of parabolas is vital when interpreting the graph of our function \( f(x) = x^2 + 3x - 2 \). In this instance, since the coefficient of \( x^2 \) is positive, the parabola opens upwards, indicating a minimum point at its vertex.