Problem 4 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The roots of the equation are 0 and -7.

Step 1: Understand the Equation

The given equation is \(-x^2 -7x = 0\). This is a quadratic equation which can be rewritten in the standard form by multiplying through by -1 to give \(x^2 + 7x = 0\). We will solve this equation by graphing.

Step 2: Express as Function

Convert the equation into a function form: \( f(x) = x^2 + 7x \). This function represents a parabola opening upwards since the coefficient of \(x^2\) is positive.

Step 3: Graph the Function

Plot the graph of \( f(x) = x^2 + 7x \). Mark the x-axis at points that are multiples of 1, plot points for various values of \(x\), and then draw the curve representing the parabola. The points where the parabola intersects the x-axis are the roots of the equation.

Step 4: Find Intersection Points

Observe the plotted graph to locate the points at which the curve intersects the x-axis. The function \( f(x) = x^2 + 7x \) intersects the x-axis where \( f(x) = 0 \). These x-intercepts are the solutions to the equation.

Step 5: Identify the Exact Roots

From the graph, identify the exact x-coordinates of the intersection points. The roots of the equation \( x^2 + 7x = 0 \) are where the graph intersects the x-axis. Usually, after graphing we can find exact roots which are \(-7\) and \(0\).

Step 6: Check Your Work (Optional)

To ensure the accuracy, verify by substituting the roots \(-7\) and \(0\) back into the original equation \(-x^2 - 7x = 0\) to check they satisfy it: both should return a value of 0.

Key Concepts

Graphing Functions

Graphing a function is visualizing it by plotting its equation on a coordinate system. For quadratic equations like \( f(x) = x^2 + 7x \), graphing helps to visualize the parabola. To graph this function, you can start by creating a table of values.
Select different values of \(x\) and calculate their corresponding \(f(x)\) values. It helps to choose both negative and positive values for a complete picture.

• For example, if \(x = 0\), then \(f(x) = 0^2 + 7 \cdot 0 = 0\).
• If \(x = -1\), then \(f(x) = (-1)^2 + 7(-1) = -6\).
• If \(x = -7\), then \(f(x) = (-7)^2 + 7(-7) = 0\).

Once you've calculated several points, you can plot them on the graph and connect them in a smooth curve. The shape will be a parabola opening upwards, as the \(x^2\) term is positive. This visual representation makes identifying the x-intercepts straightforward.

X-Intercepts

An x-intercept is a point on the graph where the function crosses the x-axis. For any point that lies on the x-axis, the value of \(y\) or \(f(x)\) is zero. Thus, to find the x-intercepts, you need to solve the equation \(f(x) = 0\).
In the given function \(f(x) = x^2 + 7x\), setting the equation equal to zero will allow us to find the x-intercepts. Factoring the quadratic, we have:\[x(x+7) = 0\]
This gives us two potential values for \(x\): either \(x = 0\) or \(x + 7 = 0\) which simplifies to \(x = -7\). Thus, the x-intercepts of the equation are \(x = -7\) and \(x = 0\). These intercepts are the roots of the equation and marking them accurately on your graph is essential for a correct graph representation.

Solving Equations by Graphing

Solving equations by graphing involves drawing the respective function on a graph and identifying where it intersects the x-axis. This method is particularly useful when dealing with quadratic equations, where factoring or using the quadratic formula might be cumbersome or unnecessary.
With our function \(f(x) = x^2 + 7x\), graphing involves following these steps to find the roots visually:

• First, convert any given quadratic equation to its standard function form as we did with \(-x^2 - 7x = 0\) to \(f(x) = x^2 + 7x\).
• Graph the function: decide on a range of x-values, calculate corresponding f(x) values, plot them, and draw the curve.
• Look for points where the curve intersects the x-axis; these intersections point to the equation's roots, where \(f(x) = 0\).

Identifying these intersecting points gives the solutions to the equation. This visual method not only confirms theoretical solutions but also provides a better understanding of the nature of the function, including identifying whether they are integers or lie between integers.