Problem 7 - 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 root is exactly at \(x = 7\).

Step 1: Rearrange the Equation

First, observe the equation given: \[-14x + x^2 + 49 = 0\]. We can rewrite it as follows: \[x^2 - 14x + 49 = 0\].This form is useful for recognizing that this is a quadratic equation.

Step 2: Identify the Parabola

The equation \[x^2 - 14x + 49 = 0\] is a quadratic equation. In standard form, the parabola can be expressed as \[y = x^2 - 14x + 49\].This allows us to graph the function as a parabola.

Step 3: Graph the Parabola

Plot the parabola given by the equation \[y = x^2 - 14x + 49\] on the graph. Note that the vertex of the parabola can be found using the vertex formula for a quadratic equation, \( x = \frac{-b}{2a} \), which gives \( x = \frac{14}{2} = 7 \). Thus, the vertex is at (7, 0).

Step 4: Determine the Roots

Since the vertex is on the x-axis at \((7, 0)\), this means that the parabola touches the x-axis at this point and does not cross it. This indicates that there is exactly one root, which is \(x = 7\).

Step 5: Conclude the Roots

Since the parabola only touches the x-axis at \(x = 7\) and does not cross it, we conclude that the exact root is \(x = 7\) and there are no other integer roots between which the roots are located.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. It is defined by a quadratic function, which has the form \( y = ax^2 + bx + c \). The direction in which the parabola opens depends on the coefficient \( a \). For example:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Understanding the shape and direction of a parabola helps in graphing the quadratic function efficiently. Parabolas have several important features, such as the vertex, axis of symmetry, and the roots of the equation, all of which can provide insights into the behavior of the graph.

Vertex Form

The vertex form of a quadratic equation is a powerful way to quickly understand the properties of a parabola. The vertex form is expressed as:\[y = a(x-h)^2 + k\]Here,

• \( (h, k) \) is the vertex of the parabola.
• \( x = h \) is the axis of symmetry.
• \( a \) determines the direction and the "width" of the opening.

Using the vertex form provides a straightforward method to graph the parabola, as the coordinates of the vertex \( (h, k) \) can be directly read off from the equation. For instance, rewriting the given equation \( x^2 - 14x + 49 \) into vertex form could simplify the analysis of the parabola's properties, providing more insight into its behavior on the graph.

Graphing Quadratics

Graphing quadratics is an effective visual method to find the roots of a quadratic equation. Here, you transform the equation into a parabolic shape on a graph. For example, when you have the equation \( y = x^2 - 14x + 49 \), you:

• Identify the vertex using \( x = \frac{-b}{2a} \). In this case, it is \( x = 7 \).
• Plot the vertex on the graph, which gives you an important reference point.
• Draw the symmetrical shape that extends from the vertex following the values decided by \( a \).

When graphing a quadratic, it's crucial to check whether and where the parabola intersects the x-axis. In this equation, the parabola touches the x-axis at \( x = 7 \), indicating that this is the only root, as it doesn’t cross the x-axis at any other point. The graphical representation of quadratics makes it easier to understand where the solutions to the equation lie in terms of real numbers.