What Is The Solution Set Of The Equation \[$(x-2)(x-a)=0\$\]?

Feb 27, 2025 by ADMIN 62 views

What is the Solution Set of the Equation ${$ (x-2)(x-a)=0$}$?

The solution set of an equation is the set of all possible values of the variable that satisfy the equation. In this case, we are given the equation ${$ (x-2)(x-a)=0$}$, and we need to find the solution set.

Understanding the Equation

The given equation is a quadratic equation in the form of ${$ (x-p)(x-q)=0$}$, where p and q are constants. In this case, p = 2 and q = a. The equation can be expanded as ${$ x^2 - (a+2)x + 2a = 0$}$.

Finding the Solution Set

To find the solution set, we need to find the values of x that satisfy the equation. We can do this by setting each factor equal to zero and solving for x.

Setting the first factor equal to zero, we get:

${$ x - 2 = 0$}$

Solving for x, we get:

${$ x = 2$}$

Setting the second factor equal to zero, we get:

${$ x - a = 0$}$

Solving for x, we get:

${$ x = a$}$

The Solution Set

The solution set of the equation ${$ (x-2)(x-a)=0$}$ is the set of all possible values of x that satisfy the equation. From the previous section, we found that x = 2 and x = a are the solutions to the equation.

Therefore, the solution set of the equation ${$ (x-2)(x-a)=0$}$ is:

${$ {2, a}$}$

Interpretation of the Solution Set

The solution set ${$ {2, a}$}$ means that the equation ${$ (x-2)(x-a)=0$}$ has two solutions: x = 2 and x = a. The value of a is a constant that is not specified in the equation, so it can take on any value.

Conclusion

Understanding the Graphical Representation

The equation ${$ (x-2)(x-a)=0$}$ can be represented graphically as two intersecting lines. The line x = 2 is a vertical line that intersects the x-axis at x = 2. The line x = a is a vertical line that intersects the x-axis at x = a.

The solution set ${$ {2, a}$}$ represents the points of intersection between the two lines. The point (2, 0) represents the solution x = 2, and the point (a, 0) represents the solution x = a.

Real-World Applications

The equation ${$ (x-2)(x-a)=0$}$ has many real-world applications. For example, in physics, the equation can be used to model the motion of an object that is subject to a constant force. In engineering, the equation can be used to design systems that involve multiple variables.

Solving Systems of Equations

The equation ${$ (x-2)(x-a)=0$}$ can be used to solve systems of equations. For example, if we have two equations:

${$ x - 2 = 0$}$

${$ x - a = 0$}$

We can solve the system of equations by substituting the value of x from the first equation into the second equation.

Solving for a

We can solve for a by substituting the value of x from the first equation into the second equation.

${$ x - a = 0$}$

Substituting x = 2, we get:

${ $2 - a = 0\$}$

Solving for a, we get:

${$ a = 2$}$

Conclusion

Final Thoughts

The equation ${$ (x-2)(x-a)=0$}$ is a simple quadratic equation that can be used to model many real-world phenomena. The solution set ${$ {2, a}$}$ represents the points of intersection between two lines, and it can be used to solve systems of equations.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Glossary

Solution set: The set of all possible values of the variable that satisfy the equation.
Quadratic equation: An equation of the form ${$ ax^2 + bx + c = 0$}$, where a, b, and c are constants.
Vertical line: A line that intersects the x-axis at a single point.
System of equations: A set of two or more equations that involve multiple variables.
Q&A: Understanding the Solution Set of the Equation ${$ (x-2)(x-a)=0$}$

In our previous article, we discussed the solution set of the equation ${$ (x-2)(x-a)=0$}$. In this article, we will answer some frequently asked questions about the solution set and provide additional insights into the equation.

Q: What is the solution set of the equation ${$ (x-2)(x-a)=0$}$?

A: The solution set of the equation ${$ (x-2)(x-a)=0$}$ is the set of all possible values of x that satisfy the equation. We found that the solution set is ${$ {2, a}$}$, where a is a constant that is not specified in the equation.

Q: What does the solution set ${$ {2, a}$}$ mean?

A: The solution set ${$ {2, a}$}$ means that the equation ${$ (x-2)(x-a)=0$}$ has two solutions: x = 2 and x = a. The value of a is a constant that is not specified in the equation, so it can take on any value.

Q: How do I find the solution set of the equation ${$ (x-2)(x-a)=0$}$?

A: To find the solution set, you need to set each factor equal to zero and solve for x. In this case, we set the first factor equal to zero and solved for x, and then set the second factor equal to zero and solved for x.

Q: What is the difference between the solution set and the graph of the equation?

A: The solution set is the set of all possible values of x that satisfy the equation, while the graph of the equation is a visual representation of the equation. The solution set is a set of points, while the graph is a line or curve.

Q: Can I use the solution set to solve systems of equations?

A: Yes, you can use the solution set to solve systems of equations. For example, if you have two equations:

${$ x - 2 = 0$}$

${$ x - a = 0$}$

You can solve the system of equations by substituting the value of x from the first equation into the second equation.

Q: How do I find the value of a in the solution set ${$ {2, a}$}$?

A: To find the value of a, you need to substitute the value of x from the first equation into the second equation. In this case, we substituted x = 2 into the second equation and solved for a.

Q: What are some real-world applications of the equation ${$ (x-2)(x-a)=0$}$?

A: The equation ${$ (x-2)(x-a)=0$}$ has many real-world applications, such as modeling the motion of an object that is subject to a constant force, designing systems that involve multiple variables, and solving systems of equations.

Q: Can I use the solution set to find the intersection points of two lines?

A: Yes, you can use the solution set to find the intersection points of two lines. The solution set ${$ {2, a}$}$ represents the points of intersection between two lines, and it can be used to find the intersection points.

Q: How do I graph the equation ${$ (x-2)(x-a)=0$}$?

A: To graph the equation, you need to graph the two lines x = 2 and x = a. The solution set ${$ {2, a}$}$ represents the points of intersection between the two lines.

Q: Can I use the solution set to solve quadratic equations?

A: Yes, you can use the solution set to solve quadratic equations. The solution set ${$ {2, a}$}$ represents the solutions to the quadratic equation ${$ (x-2)(x-a)=0$}$.

Q: How do I find the solution set of a quadratic equation?

A: To find the solution set of a quadratic equation, you need to set each factor equal to zero and solve for x. In this case, we set the first factor equal to zero and solved for x, and then set the second factor equal to zero and solved for x.

Q: What are some common mistakes to avoid when finding the solution set of a quadratic equation?

A: Some common mistakes to avoid when finding the solution set of a quadratic equation include:

Not setting each factor equal to zero
Not solving for x
Not checking for extraneous solutions
Not using the correct method to solve the equation

Q: Can I use the solution set to find the roots of a quadratic equation?

A: Yes, you can use the solution set to find the roots of a quadratic equation. The solution set ${$ {2, a}$}$ represents the roots of the quadratic equation ${$ (x-2)(x-a)=0$}$.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you need to set each factor equal to zero and solve for x. In this case, we set the first factor equal to zero and solved for x, and then set the second factor equal to zero and solved for x.

Q: What are some real-world applications of the roots of a quadratic equation?

A: The roots of a quadratic equation have many real-world applications, such as modeling the motion of an object that is subject to a constant force, designing systems that involve multiple variables, and solving systems of equations.

Q: Can I use the solution set to find the intersection points of two curves?

A: Yes, you can use the solution set to find the intersection points of two curves. The solution set ${$ {2, a}$}$ represents the points of intersection between two curves.

Q: How do I find the intersection points of two curves?

A: To find the intersection points of two curves, you need to set each curve equal to the other and solve for x. In this case, we set the two curves equal to each other and solved for x.

Q: What are some common mistakes to avoid when finding the intersection points of two curves?

A: Some common mistakes to avoid when finding the intersection points of two curves include:

Not setting each curve equal to the other
Not solving for x
Not checking for extraneous solutions
Not using the correct method to solve the equation

Q: Can I use the solution set to find the maximum or minimum value of a function?

A: Yes, you can use the solution set to find the maximum or minimum value of a function. The solution set ${$ {2, a}$}$ represents the points of maximum or minimum value of the function.

Q: How do I find the maximum or minimum value of a function?

A: To find the maximum or minimum value of a function, you need to find the critical points of the function and then evaluate the function at those points. In this case, we found the critical points of the function and then evaluated the function at those points.

Q: What are some real-world applications of the maximum or minimum value of a function?

A: The maximum or minimum value of a function has many real-world applications, such as modeling the motion of an object that is subject to a constant force, designing systems that involve multiple variables, and solving systems of equations.

Conclusion

In conclusion, the solution set of the equation ${$ (x-2)(x-a)=0$}$ is the set of all possible values of x that satisfy the equation. We found that the solution set is ${$ {2, a}$}$, where a is a constant that is not specified in the equation. The solution set can be used to solve systems of equations, find the intersection points of two lines, and find the maximum or minimum value of a function.