Problem 24 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 14 x+x^{2}+49=0 $$

Step-by-Step Solution

Answer

The equation \( 14x + x^2 + 49 = 0 \) has one repeated root at \( x = -7 \).

Step 1: Rearrange the Equation

The given equation is already arranged, but let's write it in standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 14 \), and \( c = 49 \). Hence, the equation is \( x^2 + 14x + 49 = 0 \).

Step 2: Identify the Function to Graph

The function to graph is the quadratic function \( f(x) = x^2 + 14x + 49 \). This will help us find the values of \( x \) where \( f(x) = 0 \).

Step 3: Determine the Vertex of the Parabola

The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 14 \), so \( x = -\frac{14}{2} = -7 \). Calculate \( y \) using \( y = (-7)^2 + 14(-7) + 49 = 0 \). So, the vertex \((-7, 0)\) is on the x-axis.

Step 4: Plot the Parabola

Using a graphing tool, plot the quadratic function \( f(x) = x^2 + 14x + 49 \). Since the vertex of the parabola is \((-7, 0)\) and \( f(x) \) is a parabola opening upwards, note that it touches the x-axis at \( x = -7 \), indicating a repeated root.

Step 5: Conclusion from the Graph

From the graph, observe that the parabola only touches the x-axis at \( x = -7 \). Therefore, there is only one root, and it is exact and repeated.

Key Concepts

Graphing Method

Graphing is an effective way to visually solve quadratic equations. To utilize the graphing method, you first transform the quadratic equation into a function format. By plotting this function on a graph, you can observe where it intersects the x-axis, if at all. These intersection points are the solutions or "roots" of the equation. For example, with the equation given, we graph the function \( f(x) = x^2 + 14x + 49 \). By observing the graph, we see that it touches the x-axis at the point corresponding to the root of the equation.

• To graph efficiently, identify key points such as the vertex and y-intercept.
• The graph should be a smooth curve, opening upwards if the coefficient of \( x^2 \) is positive, or downwards if negative.

By graphing, you can quickly identify real number roots or determine the absence of real solutions if the parabola does not intersect the x-axis.

Standard Form

In algebra, the standard form of a quadratic equation is crucial for solving and interpreting. Standard form is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

• Consistently automating the equation into this form simplifies many solving methods, like factoring or using the quadratic formula.
• It also aids in identifying key features of the parabola, such as the direction it opens.

In our case, the equation \( 14x + x^2 + 49 = 0 \) is rearranged to fit the standard format as \( x^2 + 14x + 49 = 0 \). This conversion is instrumental in further analysis, including vertex finding and graph plotting.

Vertex of a Parabola

The vertex of a parabola is a vital point that represents the maximum or minimum value of the function. It serves as the turning point of the graph. To find the vertex's x-coordinate in a quadratic equation in standard form, use the formula\[ x = -\frac{b}{2a} \]. Using our example, with \( a = 1 \) and \( b = 14 \), the x-coordinate is calculated as \(-7\). To find the y-coordinate, substitute \( x = -7 \) back into the equation: \( y = (-7)^2 + 14(-7) + 49 = 0 \). Thus, the vertex \((-7, 0)\) indicates that the parabola merely touches the x-axis, pinpointing a repeated root. This point is key to graphing and understanding the symmetry of the quadratic function.

• The vertex also helps to determine if the parabola has a maximum or minimum point, crucial for optimization problems.

Repeated Root

A repeated root in a quadratic equation occurs when the parabola touches the x-axis at exactly one point. This means the root is a "double root", showing in the solution of the equation as a single repeated value. In our example: \( x^2 + 14x + 49 = 0 \), the graph intersects the x-axis at \( x = -7 \). Since the parabola only touches the axis at this point, \( x = -7 \) becomes the repeated root.

• This situation occurs when the discriminant \(( b^2 - 4ac)\) equals zero. For our equation, \( (14)^2 - 4(1)(49) = 0 \).
• The graph verifies this as there is only one solution due to the tangent point.

Knowing about repeated roots is important, as it affects understanding of the curve's behavior and intersection properties. It confirms whether a quadratic equation has one distinct solution instead of the usual two.