Problem 5 - Algebra 2

Question

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}-2 x-24=0 $$

Step-by-Step Solution

Answer

Graph shows roots at \(-4\) and \(6\) or between these integers.

Step 1: Understand the Problem

We need to solve the quadratic equation \(x^2 - 2x - 24 = 0\) by graphing. This involves finding the points where the graph of the quadratic equation intersects the x-axis.

Step 2: Graph the Equation

Graph the equation \(y = x^2 - 2x - 24\) using a graphing calculator or software. The graph will be a parabola opening upwards because the coefficient of \(x^2\) is positive.

Step 3: Identify the x-Intercepts

On the graph, look for the points where the parabola crosses the x-axis. These are the solutions to the equation because they are the values of \(x\) where \(y = 0\).

Step 4: Analyze the Graph

If the graph crosses exactly at integer points, those are your roots. If it does not cross exactly at integers, identify the two consecutive integers between which the roots lie.

Key Concepts

Graphing Parabolas

A parabola is the graph of a quadratic function, which is an equation of the form \(y = ax^2 + bx + c\). To graph a parabola, it helps to understand its general shape and features.

• The leading coefficient \(a\) tells us whether the parabola opens upwards or downwards. If \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.
• The vertex is the highest or lowest point on the graph, depending on the direction it opens.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

When graphing a parabola like \(y = x^2 - 2x - 24\), you may use a graphing calculator or draw it by hand. First, identify the vertex using the formula \(x = -\frac{b}{2a}\), then plot a few points on either side of the vertex to reflect its symmetrical nature. By visualizing a parabola's graph, you can understand its key features, like where it will intersect the x-axis.

Roots of Quadratic Equations

The roots of a quadratic equation, also known as solutions or zeros, are the x-values where the parabola intersects the x-axis. These points occur where \(y=0\). For the equation \(x^2 - 2x - 24 = 0\), finding the roots is essentially solving where this quadratic expression evaluates to zero. There are several methods to find these roots:

• Factoring: Write the quadratic in the form \((x - p)(x - q) = 0\). This requires the quadratic to be factorable.
• Quadratic Formula: Use \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) when the quadratic is not easily factorable.
• Completing the Square: Rewrite the quadratic so that it becomes a perfect square trinomial.

In our given equation, notice the coefficients are integers, making it a good candidate for factoring or graphing. By finding roots, you're essentially identifying the x-values where the parabola cuts through the x-axis, representing the solutions of the quadratic equation.

X-Intercepts of a Parabola

X-intercepts are crucial when graphing quadratic equations because they inform us about the points at which the graph crosses the x-axis. These points, also known as the roots, zeros, or solutions of the equation, occur when \(y = 0\). For a parabola described by the equation \(y = x^2 - 2x - 24\), solving \(x^2 - 2x - 24 = 0\) gives us the x-intercepts. Graphically, these appear as the points where the curve meets the x-axis. To locate these intercepts:

• Graph the equation using a graphing tool and visually identify the intersections.
• If intersections are at exact coordinates, those are the precise roots.
• If the graph intersects between integers, identify the two integers between which each root lies。

Understanding x-intercepts provides insights into the behavior of the graph and helps solve the quadratic equation by showing the values at which the function equals zero. This visual representation plays a vital role in both solving equations and in comprehending the structure of parabolas.