Use The Quadratic Formula To Solve The Equation. If Necessary, Round To The Nearest Hundredth.$2a^2 - 30a + 108 = 0$A. -6, -9 B. -9, -12 C. 6, 9 D. 9, 12
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Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on using the quadratic formula to solve a quadratic equation, and we will provide a step-by-step guide on how to do it. We will also discuss the importance of rounding to the nearest hundredth when necessary.
What is the Quadratic Formula?
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is as follows:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
How to Use the Quadratic Formula
To use the quadratic formula, we need to identify the values of a, b, and c in the quadratic equation. In our example, the quadratic equation is 2a^2 - 30a + 108 = 0. We can identify the values of a, b, and c as follows:
a = 2 b = -30 c = 108
Now that we have identified the values of a, b, and c, we can plug them into the quadratic formula:
x = (30 ± √((-30)^2 - 4(2)(108))) / 2(2)
Simplifying the Quadratic Formula
To simplify the quadratic formula, we need to evaluate the expression inside the square root:
(-30)^2 = 900 4(2)(108) = 864
Now we can plug these values back into the quadratic formula:
x = (30 ± √(900 - 864)) / 4
x = (30 ± √36) / 4
x = (30 ± 6) / 4
Solving for x
Now that we have simplified the quadratic formula, we can solve for x by evaluating the two possible values:
x = (30 + 6) / 4 = 36 / 4 = 9 x = (30 - 6) / 4 = 24 / 4 = 6
Rounding to the Nearest Hundredth
Since we are asked to round to the nearest hundredth, we need to round our solutions to two decimal places. However, in this case, our solutions are integers, so we do not need to round.
Conclusion
In conclusion, we have used the quadratic formula to solve the quadratic equation 2a^2 - 30a + 108 = 0. We have identified the values of a, b, and c, plugged them into the quadratic formula, simplified the formula, and solved for x. We have also discussed the importance of rounding to the nearest hundredth when necessary.
Answer
The final answer is: 6, 9
Discussion
The quadratic formula is a powerful tool for solving quadratic equations. However, it can be complex and difficult to use, especially for students who are new to mathematics. In this article, we have provided a step-by-step guide on how to use the quadratic formula, and we have discussed the importance of rounding to the nearest hundredth when necessary.
Why is the Quadratic Formula Important?
The quadratic formula is an important tool in mathematics because it provides a way to solve quadratic equations. Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. The quadratic formula is used in a wide range of applications, including physics, engineering, and economics.
How to Use the Quadratic Formula in Real-World Applications
The quadratic formula can be used in a wide range of real-world applications, including:
- Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
- Engineering: The quadratic formula is used to solve problems involving design, such as the design of a bridge.
- Economics: The quadratic formula is used to solve problems involving optimization, such as the maximization of profit.
Common Mistakes to Avoid When Using the Quadratic Formula
When using the quadratic formula, there are several common mistakes to avoid, including:
- Incorrectly identifying the values of a, b, and c: Make sure to identify the values of a, b, and c correctly before plugging them into the quadratic formula.
- Not simplifying the quadratic formula: Make sure to simplify the quadratic formula before solving for x.
- Not rounding to the nearest hundredth: Make sure to round your solutions to the nearest hundredth when necessary.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is used in a wide range of applications, including physics, engineering, and economics. However, it can be complex and difficult to use, especially for students who are new to mathematics. By following the step-by-step guide provided in this article, students can learn how to use the quadratic formula and solve quadratic equations with confidence.
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Introduction
The quadratic formula is a powerful tool for solving quadratic equations, but it can be complex and difficult to use, especially for students who are new to mathematics. In this article, we will answer some of the most frequently asked questions about the quadratic formula, including how to use it, common mistakes to avoid, and real-world applications.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is as follows:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you plug these values into the quadratic formula and simplify the expression inside the square root. Finally, you solve for x by evaluating the two possible values.
Q: What are the common mistakes to avoid when using the quadratic formula?
A: There are several common mistakes to avoid when using the quadratic formula, including:
- Incorrectly identifying the values of a, b, and c: Make sure to identify the values of a, b, and c correctly before plugging them into the quadratic formula.
- Not simplifying the quadratic formula: Make sure to simplify the quadratic formula before solving for x.
- Not rounding to the nearest hundredth: Make sure to round your solutions to the nearest hundredth when necessary.
Q: How do I apply the quadratic formula in real-world situations?
A: The quadratic formula can be applied in a wide range of real-world situations, including:
- Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
- Engineering: The quadratic formula is used to solve problems involving design, such as the design of a bridge.
- Economics: The quadratic formula is used to solve problems involving optimization, such as the maximization of profit.
Q: What are some examples of quadratic equations that can be solved using the quadratic formula?
A: Some examples of quadratic equations that can be solved using the quadratic formula include:
- x^2 + 5x + 6 = 0
- 2x^2 - 3x - 1 = 0
- x^2 - 4x + 4 = 0
Q: How do I check my solutions to a quadratic equation?
A: To check your solutions to a quadratic equation, you can plug the solutions back into the original equation and verify that they are true. You can also use the quadratic formula to check your solutions.
Q: What are some tips for mastering the quadratic formula?
A: Some tips for mastering the quadratic formula include:
- Practice, practice, practice: The more you practice using the quadratic formula, the more comfortable you will become with it.
- Understand the concept: Make sure you understand the concept behind the quadratic formula and how it works.
- Use online resources: There are many online resources available that can help you learn and practice using the quadratic formula.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, but it can be complex and difficult to use, especially for students who are new to mathematics. By following the step-by-step guide provided in this article and answering some of the most frequently asked questions about the quadratic formula, students can learn how to use the quadratic formula and solve quadratic equations with confidence.
Additional Resources
- Quadratic Formula Calculator: A calculator that can help you solve quadratic equations using the quadratic formula.
- Quadratic Formula Worksheet: A worksheet that provides practice problems for solving quadratic equations using the quadratic formula.
- Quadratic Formula Video Tutorial: A video tutorial that provides a step-by-step guide on how to use the quadratic formula.
Final Thoughts
The quadratic formula is a powerful tool for solving quadratic equations, and with practice and patience, anyone can master it. By following the step-by-step guide provided in this article and answering some of the most frequently asked questions about the quadratic formula, students can learn how to use the quadratic formula and solve quadratic equations with confidence.