Solve For { X $}$ In The Equation { X^2 - 5x = -6 $}$.A) { {3, 2}$}$B) { {-8}$}$C) { {-3, -2}$}$D) { {-5, 4}$}$
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, { x^2 - 5x = -6 $}$, and explore the different methods and techniques used to find the solutions.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is { ax^2 + bx + c = 0 $}$, where a, b, and c are constants. In our equation, { x^2 - 5x = -6 $}$, we can rewrite it in the standard form as { x^2 - 5x + 6 = 0 $}$.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form { ax^2 + bx + c = 0 $}$, the solutions are given by:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
In our equation, { x^2 - 5x + 6 = 0 $}$, we have a = 1, b = -5, and c = 6. Plugging these values into the quadratic formula, we get:
{ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} $}$
Simplifying the expression, we get:
{ x = \frac{5 \pm \sqrt{25 - 24}}{2} $}$
{ x = \frac{5 \pm \sqrt{1}}{2} $}$
{ x = \frac{5 \pm 1}{2} $}$
This gives us two possible solutions:
{ x = \frac{5 + 1}{2} = 3 $}$
{ x = \frac{5 - 1}{2} = 2 $}$
Verifying the Solutions
To verify the solutions, we can plug them back into the original equation. Let's start with x = 3:
{ (3)^2 - 5(3) = 9 - 15 = -6 $}$
This confirms that x = 3 is a solution to the equation. Now, let's try x = 2:
{ (2)^2 - 5(2) = 4 - 10 = -6 $}$
This also confirms that x = 2 is a solution to the equation.
Conclusion
In this article, we solved the quadratic equation { x^2 - 5x = -6 $}$ using the quadratic formula. We found two solutions, x = 3 and x = 2, and verified them by plugging them back into the original equation. This demonstrates the power of the quadratic formula in solving quadratic equations.
Answer
The correct answer is:
{ {3, 2} $}$
This corresponds to option A) { {3, 2} $}$.
Additional Tips and Resources
- To solve quadratic equations, you can use the quadratic formula or factorization.
- The quadratic formula is a powerful tool for solving quadratic equations, but it can be complex to use.
- Factorization is a simpler method for solving quadratic equations, but it may not always be possible.
- You can use online resources, such as calculators or math software, to help you solve quadratic equations.
- Practice solving quadratic equations to become more comfortable with the different methods and techniques.
Common Mistakes to Avoid
- Make sure to plug the solutions back into the original equation to verify them.
- Be careful when simplifying the expression in the quadratic formula.
- Don't forget to include the ± symbol when using the quadratic formula.
- Make sure to use the correct values for a, b, and c in the quadratic formula.
Real-World Applications
- Quadratic equations have many real-world applications, such as modeling population growth, projectile motion, and electrical circuits.
- Solving quadratic equations is a crucial skill for professionals in fields such as engineering, physics, and computer science.
- Quadratic equations can be used to model complex systems and make predictions about future behavior.
Conclusion
Frequently Asked Questions
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is { ax^2 + bx + c = 0 $}$, where a, b, and c are constants.
Q: How do I solve a quadratic equation?
A: There are several methods for solving quadratic equations, including the quadratic formula, factorization, and graphing. The quadratic formula is a powerful tool for solving quadratic equations, but it can be complex to use. Factorization is a simpler method for solving quadratic equations, but it may not always be possible.
Q: What is the quadratic formula?
A: The quadratic formula is a formula for solving quadratic equations. It states that for an equation in the form { ax^2 + bx + c = 0 $}$, the solutions are given by:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c from the quadratic equation into the formula. Then, simplify the expression and solve for x.
Q: What are the solutions to a quadratic equation?
A: The solutions to a quadratic equation are the values of x that satisfy the equation. In other words, they are the values of x that make the equation true.
Q: How do I verify the solutions to a quadratic equation?
A: To verify the solutions to a quadratic equation, you need to plug them back into the original equation. If the equation is true for the solution, then it is a valid solution.
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations include:
- Not plugging the solutions back into the original equation to verify them
- Not simplifying the expression in the quadratic formula correctly
- Not including the ± symbol when using the quadratic formula
- Not using the correct values for a, b, and c in the quadratic formula
Q: What are some real-world applications of quadratic equations?
A: Quadratic equations have many real-world applications, such as modeling population growth, projectile motion, and electrical circuits. Solving quadratic equations is a crucial skill for professionals in fields such as engineering, physics, and computer science.
Q: How can I practice solving quadratic equations?
A: You can practice solving quadratic equations by working through examples and exercises in a textbook or online resource. You can also use online calculators or math software to help you solve quadratic equations.
Q: What are some additional resources for learning about quadratic equations?
A: Some additional resources for learning about quadratic equations include:
- Online tutorials and videos
- Math textbooks and workbooks
- Online calculators and math software
- Math classes or tutoring sessions
Conclusion
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations and become more comfortable with the different methods and techniques. Remember to practice solving quadratic equations to become more proficient, and don't be afraid to ask for help if you need it.