Solve The Equation Using The Quadratic Formula: 6 X 2 − 2 X = 8 6x^2 - 2x = 8 6 X 2 − 2 X = 8 A. X = 3 4 , − 1 X = \frac{3}{4}, -1 X = 4 3 , − 1 B. X = 3 4 , 1 X = \frac{3}{4}, 1 X = 4 3 , 1 C. X = 3 4 , 0 X = \frac{3}{4}, 0 X = 4 3 , 0 D. X = 4 3 , − 1 X = \frac{4}{3}, -1 X = 3 4 , − 1 Please Select The Best Answer From The Choices
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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 delve into the world of quadratic equations and explore the quadratic formula, a powerful tool for solving these equations. We will also apply the quadratic formula to a specific equation and analyze the results.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
The Quadratic Formula
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the constants from the quadratic equation.
Applying the Quadratic Formula
Now that we have the quadratic formula, let's apply it to the given equation:
6x^2 - 2x = 8
First, we need to rewrite the equation in the standard form of a quadratic equation:
6x^2 - 2x - 8 = 0
Now, we can identify the values of a, b, and c:
a = 6, b = -2, and c = -8
Substituting these values into the quadratic formula, we get:
x = (2 ± √((-2)^2 - 4(6)(-8))) / 2(6)
x = (2 ± √(4 + 192)) / 12
x = (2 ± √196) / 12
x = (2 ± 14) / 12
Solving for x
Now that we have the quadratic formula applied to the equation, we can solve for x. We have two possible solutions:
x = (2 + 14) / 12
x = 16 / 12
x = 4/3
x = (2 - 14) / 12
x = -12 / 12
x = -1
Analyzing the Results
We have two possible solutions for x: 4/3 and -1. To determine which solution is correct, we need to substitute each value back into the original equation and check if it is true.
Substituting x = 4/3 into the original equation, we get:
6(4/3)^2 - 2(4/3) = 8
64/9 - 8/3 = 8
64/9 - 24/9 = 8
40/9 ≠ 8
This is not true, so x = 4/3 is not a solution.
Substituting x = -1 into the original equation, we get:
6(-1)^2 - 2(-1) = 8
6 - 2 = 8
4 ≠ 8
This is not true, so x = -1 is not a solution.
Conclusion
In this article, we applied the quadratic formula to a specific equation and analyzed the results. We found that the solutions to the equation are x = 3/4 and x = -1. However, upon further analysis, we found that x = 3/4 is the correct solution.
Final Answer
The final answer is:
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Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In our previous article, we explored the quadratic formula and applied it to a specific equation. In this article, we will provide a Q&A guide to help you better understand quadratic equations and the quadratic formula.
Q&A
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 (usually x) is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the constants from the quadratic equation.
Q: How do I apply the quadratic formula?
A: To apply the quadratic formula, you need to identify the values of a, b, and c from the quadratic equation. Then, substitute these values into the quadratic formula and simplify the expression.
Q: What are the steps to solve a quadratic equation using the quadratic formula?
A: The steps to solve a quadratic equation using the quadratic formula are:
- Identify the values of a, b, and c from the quadratic equation.
- Substitute these values into the quadratic formula.
- Simplify the expression.
- Solve for x.
Q: What are the possible solutions to a quadratic equation?
A: The possible solutions to a quadratic equation are given by the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
There are two possible solutions: x = (-b + √(b^2 - 4ac)) / 2a and x = (-b - √(b^2 - 4ac)) / 2a.
Q: How do I determine which solution is correct?
A: To determine which solution is correct, you need to substitute each value back into the original equation and check if it is true.
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations are:
- Not identifying the values of a, b, and c correctly.
- Not substituting the values into the quadratic formula correctly.
- Not simplifying the expression correctly.
- Not checking if the solutions are true.
Tips and Tricks
Tip 1: Make sure to identify the values of a, b, and c correctly.
When solving a quadratic equation, it is essential to identify the values of a, b, and c correctly. This will ensure that you substitute the correct values into the quadratic formula.
Tip 2: Simplify the expression carefully.
When simplifying the expression, make sure to follow the order of operations (PEMDAS). This will ensure that you get the correct solution.
Tip 3: Check if the solutions are true.
After finding the solutions, make sure to check if they are true by substituting them back into the original equation.
Conclusion
In this article, we provided a Q&A guide to help you better understand quadratic equations and the quadratic formula. We covered topics such as the definition of a quadratic equation, the quadratic formula, and how to apply it. We also provided tips and tricks to help you avoid common mistakes when solving quadratic equations.
Final Answer
The final answer is: