Find Other Zero Of Quadratic Equation With Axis Of Symmetry
Understanding the properties of quadratic equations is crucial in mathematics. One such property involves the axis of symmetry, which plays a vital role in determining the relationship between the zeroes of the equation. In this article, we will explore how to find the other zero of a quadratic equation when given the axis of symmetry and one zero. Specifically, we will address the problem where the axis of symmetry is x = -3 and one zero is 4. This comprehensive guide will delve into the underlying concepts, provide a step-by-step solution, and offer additional insights to enhance your understanding of quadratic equations.
Understanding Quadratic Equations and Their Properties
Before we dive into solving the problem, let's establish a solid foundation by revisiting the fundamentals of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form:
ax² + bx + c = 0
where a, b, and c are constants, and a ≠ 0. The solutions to this equation are called the roots or zeroes of the quadratic equation, which are the x-values that satisfy the equation. A quadratic equation can have at most two distinct real roots, one repeated real root, or two complex roots. These roots correspond to the x-intercepts of the parabola represented by the quadratic equation when graphed on the Cartesian plane.
The Axis of Symmetry: A Key Concept
The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. This line passes through the vertex of the parabola, which is the point where the parabola changes direction (either the minimum or maximum point). The x-coordinate of the vertex represents the equation of the axis of symmetry. For a quadratic equation in the standard form ax² + bx + c = 0, the axis of symmetry is given by the formula:
x = -b / 2a
The axis of symmetry is crucial because it tells us that the two roots of the quadratic equation are equidistant from this line. In other words, if we know the axis of symmetry and one root, we can easily find the other root by using the symmetry property.
Relationship Between Zeroes and the Axis of Symmetry
As mentioned earlier, the zeroes of a quadratic equation are symmetrically positioned around the axis of symmetry. If we denote the two zeroes as x₁ and x₂, and the axis of symmetry as x = h, then the midpoint of the zeroes will lie on the axis of symmetry. Mathematically, this can be expressed as:
(x₁ + x₂) / 2 = h
This relationship forms the basis for finding the other zero when one zero and the axis of symmetry are known. We can rearrange this formula to solve for the unknown zero:
x₂ = 2h - x₁
This equation provides a direct method to calculate the other zero, making it a powerful tool in solving quadratic equation problems.
Solving for the Other Zero: A Step-by-Step Approach
Now that we have a clear understanding of the properties of quadratic equations and the significance of the axis of symmetry, let's apply this knowledge to solve the given problem. We are given that the axis of symmetry of a quadratic equation is x = -3 and one of the zeroes is 4. Our goal is to find the other zero.
Step 1: Identify the Given Information
First, we need to clearly identify the information provided in the problem:
- Axis of symmetry: x = -3
- One zero: x₁ = 4
Step 2: Apply the Symmetry Formula
We know that the relationship between the zeroes and the axis of symmetry is given by:
x₂ = 2h - x₁
where x₂ is the other zero, h is the x-coordinate of the axis of symmetry, and x₁ is the given zero.
Step 3: Substitute the Given Values
Substitute the given values into the formula:
x₂ = 2(-3) - 4
Step 4: Simplify the Equation
Now, simplify the equation to find the value of x₂:
x₂ = -6 - 4 x₂ = -10
Therefore, the other zero of the quadratic equation is -10.
Verifying the Solution
To ensure our solution is correct, we can verify it by checking if the midpoint of the two zeroes (-10 and 4) lies on the axis of symmetry (x = -3). The midpoint is calculated as:
Midpoint = (x₁ + x₂) / 2
Substituting the values:
Midpoint = (4 + (-10)) / 2 Midpoint = -6 / 2 Midpoint = -3
The midpoint is indeed -3, which matches the x-coordinate of the axis of symmetry. This confirms that our solution, x₂ = -10, is correct.
Additional Insights and Applications
The concept of the axis of symmetry and its relationship to the zeroes of a quadratic equation is not just limited to finding missing zeroes. It has broader applications in understanding the behavior of quadratic functions and their graphs.
Graphing Quadratic Equations
The axis of symmetry helps in sketching the graph of a quadratic equation. Knowing the axis of symmetry and the vertex allows us to determine the parabola's orientation (whether it opens upwards or downwards) and its position on the coordinate plane. The zeroes, along with the vertex, provide key points for accurately plotting the parabola.
Finding the Vertex
The vertex of the parabola is the point where the axis of symmetry intersects the parabola. The x-coordinate of the vertex is the same as the axis of symmetry. To find the y-coordinate of the vertex, substitute the x-coordinate into the quadratic equation.
Real-World Applications
Quadratic equations have numerous applications in real-world scenarios, such as projectile motion, optimization problems, and engineering designs. Understanding the axis of symmetry can help in determining maximum or minimum values, optimal trajectories, and symmetrical designs.
Common Mistakes to Avoid
When solving problems involving the axis of symmetry and zeroes of quadratic equations, it's essential to avoid common mistakes that can lead to incorrect answers.
Misinterpreting the Axis of Symmetry
The axis of symmetry is a vertical line, not a point. It's crucial to understand that it is represented by an equation of the form x = h, where h is a constant.
Incorrectly Applying the Formula
Ensure that you correctly substitute the values into the formula x₂ = 2h - x₁. A common mistake is to mix up the values or apply the formula incorrectly.
Neglecting the Sign
Pay close attention to the signs of the numbers, especially when dealing with negative values. Incorrect signs can significantly alter the result.
Practice Problems
To reinforce your understanding, let's work through a few practice problems.
Problem 1
The axis of symmetry of a quadratic equation is x = 2, and one zero is -1. Find the other zero.
Solution: Using the formula x₂ = 2h - x₁, where h = 2 and x₁ = -1:
x₂ = 2(2) - (-1) x₂ = 4 + 1 x₂ = 5
The other zero is 5.
Problem 2
A quadratic equation has an axis of symmetry at x = -4, and one zero is -7. What is the other zero?
Solution: Using the formula x₂ = 2h - x₁, where h = -4 and x₁ = -7:
x₂ = 2(-4) - (-7) x₂ = -8 + 7 x₂ = -1
The other zero is -1.
Conclusion
In summary, finding the other zero of a quadratic equation when given the axis of symmetry and one zero is a straightforward process that relies on the symmetrical properties of parabolas. By understanding the relationship between the zeroes and the axis of symmetry, we can effectively use the formula x₂ = 2h - x₁ to solve such problems. This concept is fundamental in understanding and working with quadratic equations and their applications. By mastering this technique, you can enhance your problem-solving skills and gain a deeper understanding of quadratic functions. Remember to always verify your solutions and practice regularly to solidify your knowledge. The axis of symmetry is a powerful tool in the realm of quadratic equations, and its understanding opens doors to a more profound appreciation of mathematical concepts and their real-world applications. We hope this guide has provided you with a clear and comprehensive understanding of how to find the other zero of a quadratic equation, empowering you to tackle similar problems with confidence. Continue exploring the fascinating world of mathematics, and you'll discover the beauty and practicality it holds.
