Intersection Points Of Linear And Quadratic Equations

Jul 11, 2025 by ADMIN 54 views

In the realm of mathematics, a fundamental concept involves finding the intersection points between different types of equations. Specifically, understanding how linear and quadratic equations interact is crucial. This article will delve into the method of determining the intersection points between a linear equation, represented as $y = 2x + 5$ , and a quadratic equation, given by $y = x^2 - 2x - 7$ . We will explore the algebraic approach to solve this problem and interpret the results geometrically. The process involves setting the two equations equal to each other, simplifying the resulting quadratic equation, and analyzing the discriminant to determine the number of intersection points. Understanding this process is vital for various applications in mathematics, physics, and engineering, where the intersection of curves plays a significant role.

To begin, let's restate the problem clearly. We are given two equations: a linear equation $y = 2x + 5$ and a quadratic equation $y = x^2 - 2x - 7$ . Our objective is to determine the number of points at which the graphs of these two equations intersect. This means we need to find the number of solutions for $x$ and $y$ that satisfy both equations simultaneously. Geometrically, this corresponds to the number of points where the line and the parabola intersect on a coordinate plane. This question tests our understanding of algebraic manipulation and the properties of quadratic equations. The solution involves equating the two expressions for $y$ , forming a quadratic equation, and then using the discriminant to ascertain the number of real roots, which directly corresponds to the number of intersection points.

The core of our solution lies in an algebraic approach. To find the points of intersection, we set the two equations equal to each other:

$2x + 5 = x^2 - 2x - 7$

This step is crucial as it allows us to find the x-coordinates where the y-values of both functions are the same, hence the points of intersection. Now, we need to rearrange this equation into the standard quadratic form, which is $ax^2 + bx + c = 0$ . Subtracting $2x$ and $5$ from both sides, we get:

$0 = x^2 - 2x - 7 - 2x - 5$

Simplifying the equation, we combine like terms:

$0 = x^2 - 4x - 12$

Now we have a standard quadratic equation in the form $ax^2 + bx + c = 0$ , where $a = 1$ , $b = -4$ , and $c = -12$ . This equation represents the x-coordinates of the intersection points. To find the number of solutions, we will use the discriminant.

The discriminant is a powerful tool in determining the nature and number of roots of a quadratic equation. For a quadratic equation of the form $ax^2 + bx + c = 0$ , the discriminant, denoted as $D$ , is given by the formula:

$D = b^2 - 4ac$

The discriminant provides valuable information about the roots of the quadratic equation:

If $D > 0$ , the equation has two distinct real roots, meaning the graphs intersect at two points.
If $D = 0$ , the equation has one real root (a repeated root), meaning the graphs intersect at one point (tangency).
If $D < 0$ , the equation has no real roots, meaning the graphs do not intersect.

In our case, we have $a = 1$ , $b = -4$ , and $c = -12$ . Plugging these values into the discriminant formula, we get:

$D = (-4)^2 - 4(1)(-12)$

$D = 16 + 48$

$D = 64$

Since $D = 64 > 0$ , the quadratic equation has two distinct real roots. This means the graphs of the given linear and quadratic equations intersect at two points.

In conclusion, by setting the equations $y = 2x + 5$ and $y = x^2 - 2x - 7$ equal to each other and analyzing the resulting quadratic equation, we have determined that the graphs intersect at two points. This was achieved by calculating the discriminant, which was found to be 64, indicating two distinct real roots. Therefore, the correct answer is C. 2. This exercise demonstrates the importance of algebraic techniques in solving geometric problems and highlights the significance of the discriminant in understanding the nature of quadratic equations.

When faced with the question of how many points the graphs of the equations $y = 2x + 5$ and $y = x^2 - 2x - 7$ intersect, several approaches can be taken. However, the most efficient and accurate method involves a combination of algebraic manipulation and the application of the discriminant. The correct answer hinges on a clear understanding of how linear and quadratic equations interact graphically and algebraically. Before diving into the solution process, it's essential to recognize that the points of intersection represent the solutions that satisfy both equations simultaneously. This means that at the points of intersection, the x and y values for both equations are equal. This concept forms the basis for our algebraic approach.

The initial step in determining the number of intersection points is to set the two equations equal to each other. This allows us to create a single equation in terms of x, which can then be solved to find the x-coordinates of the intersection points. By equating the expressions for y, we obtain:

$2x + 5 = x^2 - 2x - 7$

This equation represents the condition where the y-values of the linear and quadratic functions are the same. The next step involves rearranging this equation into the standard quadratic form, $ax^2 + bx + c = 0$ . This is a crucial step as it allows us to apply the discriminant, a powerful tool for determining the nature and number of roots of a quadratic equation. Subtracting $2x$ and $5$ from both sides of the equation, we get:

$0 = x^2 - 2x - 7 - 2x - 5$

Simplifying the equation by combining like terms, we arrive at:

$0 = x^2 - 4x - 12$

Now, we have a quadratic equation in the standard form, where $a = 1$ , $b = -4$ , and $c = -12$ . The next crucial step is to calculate the discriminant. The discriminant, denoted as $D$ , is given by the formula:

$D = b^2 - 4ac$

The value of the discriminant provides critical information about the number of real roots of the quadratic equation, which directly corresponds to the number of intersection points between the two graphs. If $D > 0$ , the equation has two distinct real roots, indicating two intersection points. If $D = 0$ , the equation has one real root (a repeated root), indicating one intersection point (tangency). If $D < 0$ , the equation has no real roots, indicating no intersection points. In our case, we have $a = 1$ , $b = -4$ , and $c = -12$ . Plugging these values into the discriminant formula, we get:

$D = (-4)^2 - 4(1)(-12)$

$D = 16 + 48$

$D = 64$

Since $D = 64 > 0$ , the quadratic equation has two distinct real roots. This means the graphs of the given linear and quadratic equations intersect at two points. Therefore, the correct answer is C. 2.

The solution process for determining the intersection points between the graphs of $y = 2x + 5$ and $y = x^2 - 2x - 7$ involves a systematic approach that combines algebraic manipulation with the application of the discriminant. Each step is crucial and builds upon the previous one to arrive at the correct conclusion. The first step in the process is to recognize that the points of intersection occur where the y-values of both equations are equal. This fundamental understanding allows us to set the two equations equal to each other, creating a single equation in terms of x.

By equating the expressions for y, we obtain:

$2x + 5 = x^2 - 2x - 7$

This equation represents the condition where the graphs of the linear and quadratic functions intersect. The next step is to rearrange this equation into the standard quadratic form, $ax^2 + bx + c = 0$ . This form is essential for applying the discriminant and determining the nature and number of roots. To rearrange the equation, we subtract $2x$ and $5$ from both sides:

$0 = x^2 - 2x - 7 - 2x - 5$

Simplifying the equation by combining like terms, we get:

$0 = x^2 - 4x - 12$

$D = b^2 - 4ac$

The discriminant provides valuable information about the number of real roots of the quadratic equation. The relationship between the discriminant and the number of roots is as follows:

If $D > 0$ , the equation has two distinct real roots.
If $D = 0$ , the equation has one real root (a repeated root).
If $D < 0$ , the equation has no real roots.

In our case, we have $a = 1$ , $b = -4$ , and $c = -12$ . Plugging these values into the discriminant formula, we get:

$D = (-4)^2 - 4(1)(-12)$

$D = 16 + 48$

$D = 64$

Since $D = 64 > 0$ , the quadratic equation has two distinct real roots. This indicates that the graphs of the given linear and quadratic equations intersect at two points. Therefore, the correct answer is C. 2. This detailed explanation highlights the step-by-step process, emphasizing the importance of each stage, from setting the equations equal to each other to calculating and interpreting the discriminant. Understanding this process is crucial for solving similar problems involving the intersection of different types of equations.

Visualizing the graphs of the equations $y = 2x + 5$ and $y = x^2 - 2x - 7$ can provide a deeper understanding of why they intersect at a specific number of points. The equation $y = 2x + 5$ represents a linear function, which graphically is a straight line. The equation $y = x^2 - 2x - 7$ represents a quadratic function, which graphically is a parabola. The points of intersection between these two graphs are the points where the line and the parabola meet on the coordinate plane. To fully grasp the concept, let's consider the characteristics of each graph.

The linear equation $y = 2x + 5$ has a slope of 2 and a y-intercept of 5. This means that the line rises 2 units for every 1 unit it moves to the right, and it crosses the y-axis at the point (0, 5). The quadratic equation $y = x^2 - 2x - 7$ represents a parabola. The coefficient of the $x^2$ term is positive (1), indicating that the parabola opens upwards. To find the vertex of the parabola, we can use the formula $x = -b / (2a)$ , where $a = 1$ and $b = -2$ . Plugging in these values, we get:

$x = -(-2) / (2 * 1) = 1$

To find the y-coordinate of the vertex, we substitute $x = 1$ into the quadratic equation:

$y = (1)^2 - 2(1) - 7 = 1 - 2 - 7 = -8$

So, the vertex of the parabola is at the point (1, -8). Now that we have the vertex, we can visualize the shape and position of the parabola. The parabola opens upwards and its vertex is below the x-axis. When a line and a parabola intersect, there are three possible scenarios:

The line does not intersect the parabola at all.
The line intersects the parabola at one point (tangency).
The line intersects the parabola at two points.

In our case, we have determined algebraically that the discriminant is positive, which means there are two distinct real roots. This corresponds to the third scenario, where the line intersects the parabola at two points. Visually, this means that the straight line $y = 2x + 5$ crosses the parabola $y = x^2 - 2x - 7$ at two distinct locations on the coordinate plane. These two points are the solutions to the system of equations, and they represent the coordinates where both equations are satisfied simultaneously. Understanding the graphical representation helps to solidify the algebraic solution and provides a more intuitive understanding of the problem. The two intersection points indicate that there are two sets of x and y values that satisfy both the linear and quadratic equations, confirming our algebraic conclusion.

The discriminant plays a pivotal role in solving quadratic equations and understanding their nature. For a quadratic equation in the standard form $ax^2 + bx + c = 0$ , the discriminant, denoted as $D$ , is given by the formula:

$D = b^2 - 4ac$

The value of the discriminant provides critical information about the roots of the quadratic equation without actually solving for the roots. This is particularly useful in situations where we only need to know the number and type of roots, rather than their exact values. The discriminant allows us to determine whether the quadratic equation has:

Two distinct real roots (when $D > 0$ )
One real root (a repeated root) (when $D = 0$ )
No real roots (when $D < 0$ )

In the context of finding the intersection points between a linear equation and a quadratic equation, the discriminant is invaluable. When we set the two equations equal to each other and rearrange the resulting equation into the standard quadratic form, the discriminant helps us determine the number of intersection points. As we've seen in the problem at hand, a positive discriminant indicates two intersection points, a zero discriminant indicates one intersection point (tangency), and a negative discriminant indicates no intersection points.

The importance of the discriminant extends beyond just counting intersection points. It also provides insights into the nature of the solutions. For instance, if the discriminant is positive and a perfect square, the roots are rational numbers. If the discriminant is positive but not a perfect square, the roots are irrational numbers. This level of detail can be crucial in various mathematical and scientific applications. Moreover, the discriminant is a key concept in advanced mathematics, such as complex analysis and number theory. It is used to classify quadratic forms, analyze conic sections, and study the properties of polynomials.

In summary, the discriminant is not just a formula for determining the number of roots; it is a powerful tool for understanding the fundamental properties of quadratic equations. Its ability to provide information about the nature and type of roots makes it an indispensable concept in mathematics and related fields. In the problem of finding intersection points, the discriminant serves as a quick and efficient way to determine the number of points without the need for more complex algebraic manipulations. Understanding and applying the discriminant effectively is a key skill for anyone studying mathematics or related disciplines.