Solving Systems Of Equations Determining Real Solutions For Quadratic And Linear Systems
In the realm of mathematics, systems of equations play a pivotal role in modeling and solving real-world problems. These systems often involve multiple equations with multiple variables, and the solutions represent the points where the equations intersect or hold true simultaneously. In this article, we delve into the fascinating world of systems of equations, focusing on a specific case involving a quadratic equation and a linear equation. Our primary objective is to determine the number of solutions that such a system possesses. We will explore the underlying concepts, techniques for solving, and graphical interpretations to provide a comprehensive understanding of the topic.
To begin our exploration, let's carefully examine the given system of equations:
y = x^2 + x + 3
y = -2x - 5
The first equation, y = x^2 + x + 3, represents a quadratic equation. Quadratic equations are characterized by the presence of a squared term (x^2), which gives their graphs a distinctive parabolic shape. The coefficients of the quadratic equation, in this case, are 1 (for x^2), 1 (for x), and 3 (the constant term). The parabola opens upwards because the coefficient of the x^2 term is positive.
The second equation, y = -2x - 5, represents a linear equation. Linear equations are characterized by a constant rate of change, and their graphs are straight lines. The equation is in slope-intercept form (y = mx + b), where -2 represents the slope and -5 represents the y-intercept. The negative slope indicates that the line slopes downwards from left to right.
To determine the number of solutions the system has, we need to find the points where the parabola and the line intersect. This can be achieved by setting the two equations equal to each other:
x^2 + x + 3 = -2x - 5
Now, let's rearrange the equation to get a quadratic equation in standard form (ax^2 + bx + c = 0):
x^2 + x + 3 + 2x + 5 = 0
x^2 + 3x + 8 = 0
We now have a quadratic equation in standard form: x^2 + 3x + 8 = 0. To determine the number of real solutions, we can use the discriminant, which is a part of the quadratic formula.
The discriminant is a powerful tool that helps us determine the nature and number of solutions of a quadratic equation without actually solving for the roots. For a quadratic equation in the standard form ax^2 + bx + c = 0, the discriminant (Δ) is given by:
Δ = b^2 - 4ac
The discriminant can have three possible outcomes:
- Δ > 0: The quadratic equation has two distinct real solutions.
- Δ = 0: The quadratic equation has exactly one real solution (a repeated root).
- Δ < 0: The quadratic equation has no real solutions (two complex solutions).
In our case, the quadratic equation is x^2 + 3x + 8 = 0. Let's identify the coefficients:
- a = 1
- b = 3
- c = 8
Now, we can calculate the discriminant:
Δ = 3^2 - 4 * 1 * 8
Δ = 9 - 32
Δ = -23
Since the discriminant (Δ) is -23, which is less than 0, we can conclude that the quadratic equation x^2 + 3x + 8 = 0 has no real solutions.
To further solidify our understanding, let's consider the graphical interpretation of the system of equations. The quadratic equation y = x^2 + x + 3 represents a parabola, and the linear equation y = -2x - 5 represents a straight line. The solutions to the system of equations correspond to the points where the parabola and the line intersect.
Since we determined that the discriminant is negative, indicating no real solutions, it means that the parabola and the line do not intersect in the real coordinate plane. The parabola opens upwards, and the line has a negative slope, so they never cross each other.
In this article, we explored a system of equations consisting of a quadratic equation and a linear equation. By setting the equations equal to each other, we obtained a quadratic equation, x^2 + 3x + 8 = 0. We then used the discriminant to determine the nature and number of solutions. The discriminant, calculated as -23, was less than 0, indicating that the quadratic equation has no real solutions.
Graphically, this means that the parabola and the line do not intersect in the real coordinate plane. Therefore, the system of equations has no real solutions. This comprehensive analysis, encompassing algebraic manipulation and graphical interpretation, provides a deep understanding of the solution set for the given system of equations.
Based on our analysis, the correct answer is:
A. no real solutions
How many real solutions does the following system of equations have?
y = x^2 + x + 3
y = -2x - 5
Solving Systems of Equations Determining Real Solutions for Quadratic and Linear Systems
