Problem 23 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}-9 x=-18 $$

Step-by-Step Solution

Answer

The roots of \( x^{2}-9x=-18 \) are \( x = 3 \) and \( x = 6 \).

Step 1: Move All Terms to One Side

To solve the quadratic equation by graphing, first rewrite the equation in standard form. Move all terms to one side of the equation to get:\[ x^2 - 9x + 18 = 0 \]

Step 2: Identify the Equation to Graph

The equation \( x^2 - 9x + 18 = 0 \) is equivalent to graphing the function \( y = x^2 - 9x + 18 \). The roots of the original equation are the x-values where this graph crosses the x-axis.

Step 3: Graph the Function Using Points

Calculate key points for the graph by selecting values for \( x \) and determining \( y \):- For \( x = 0 \), \( y = 18 \).- For \( x = 3 \), \( y = 3^2 - 9 \times 3 + 18 = 0 \).- For \( x = 6 \), \( y = 6^2 - 9 \times 6 + 18 = 0 \).These calculations show that the graph crosses the x-axis at \( x = 3 \) and \( x = 6 \).

Step 4: Plot the Parabola

Plot the points calculated in Step 3 on a Cartesian plane, then draw the parabola. The points where the parabola crosses the x-axis are the roots of the equation.

Step 5: Identify the Roots

Since the parabola is plotted, the points where it intersects the x-axis give us the roots of the equation. From Step 3, we know these are at \( x = 3 \) and \( x = 6 \).

Key Concepts

Graphing

Graphing is a powerful method to visualize and solve quadratic equations like the one given in the exercise. When you graph a quadratic equation in the form of a function such as \( y = x^2 - 9x + 18 \), you are essentially sketching a curve known as a parabola. The key aspect of graphing is to look for the points where the parabola intersects the x-axis. These intersection points correspond to the roots or solutions of the quadratic equation.To graph the function effectively:

• First, convert the quadratic equation into the function form \( y = f(x) \)
• Select a range of x-values and compute their corresponding y-values to find key points
• Draw the curve using these points, ensuring it's a symmetrical U-shaped parabola

Graphing is a visual approach that gives you a clear picture of where the solutions are, even before doing any algebraic calculations.

Quadratic Function

A quadratic function is any function that can be expressed in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of the exercise, the function \( y = x^2 - 9x + 18 \) represents a quadratic equation.The most significant feature of a quadratic function is its parabolic graph, which opens upwards if \( a > 0 \) and downwards if \( a < 0 \). For \( y = x^2 - 9x + 18 \), the parabola opens upward as the coefficient of \( x^2 \) is positive (\( a = 1 \)).Important characteristics of the quadratic function include:

• The vertex of the parabola, which is the highest or lowest point
• The axis of symmetry, a vertical line passing through the vertex and splitting the parabola into two mirror images
• The intercepts where the graph crosses the axes

Understanding these aspects help in analyzing the function's behavior and solving the equation effectively.

Roots of Equations

Finding the roots of a quadratic equation involves identifying the x-values where the graph of the quadratic function intersects the x-axis. These intersections are where the graph shows \( y = 0 \).In the exercise, we determine the roots by graphing the function \( y = x^2 - 9x + 18 \), which intersects the x-axis at \( x = 3 \) and \( x = 6 \). Thus, the roots are \( x = 3 \) and \( x = 6 \).Calculating the roots can also be done algebraically using methods such as:

• Factoring the quadratic expression into \((x - 3)(x - 6)\)
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Roots of quadratic equations give us valuable information about the function, including the solutions to real-life problems modeled by these equations. Understanding how to find them graphically and algebraically enhances problem-solving skills.