Solve The Equation Using The Quadratic Formula: 25 X 2 − 169 = 0 25x^2 - 169 = 0 25 X 2 − 169 = 0 A. X = ± 13 5 X = \pm \frac{13}{5} X = ± 5 13 B. X = 13 5 , 0 X = \frac{13}{5}, 0 X = 5 13 , 0 C. X = 0 X = 0 X = 0 D. X = − 13 5 , 0 X = -\frac{13}{5}, 0 X = − 5 13 , 0
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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 explore how to solve quadratic equations using the quadratic formula. We will also provide a step-by-step guide on how to apply the formula to a given equation.
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 given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
How to Apply the Quadratic Formula
To apply the quadratic formula, we need to identify the values of a, b, and c in the given equation. In this case, we have the equation 25x^2 - 169 = 0. We can rewrite this equation in the standard form as:
25x^2 = 169
Now, we can identify the values of a, b, and c:
a = 25 b = 0 c = -169
Step 1: Plug in the Values into the Quadratic Formula
Now that we have identified the values of a, b, and c, we can plug them into the quadratic formula:
x = (-(0) ± √((0)^2 - 4(25)(-169))) / 2(25)
Step 2: Simplify the Expression
We can simplify the expression inside the square root:
x = (0 ± √(0 + 16900)) / 50
x = (0 ± √16900) / 50
x = (0 ± 130) / 50
Step 3: Solve for x
Now, we can solve for x by considering both the positive and negative cases:
x = (0 + 130) / 50 x = 130 / 50 x = 13 / 5
x = (0 - 130) / 50 x = -130 / 50 x = -13 / 5
Conclusion
Therefore, the solutions to the equation 25x^2 - 169 = 0 are x = 13/5 and x = -13/5.
Discussion
The quadratic formula is a powerful tool for solving quadratic equations. It provides a systematic approach to finding the solutions to a quadratic equation, and it can be applied to a wide range of problems. In this article, we have shown how to apply the quadratic formula to a given equation and how to simplify the expression to find the solutions.
Common Mistakes to Avoid
When applying the quadratic formula, there are several common mistakes to avoid:
- Incorrectly identifying the values of a, b, and c: Make sure to carefully identify the values of a, b, and c in the given equation.
- Not simplifying the expression: Make sure to simplify the expression inside the square root to avoid unnecessary calculations.
- Not considering both the positive and negative cases: Make sure to consider both the positive and negative cases when solving for x.
Real-World Applications
The quadratic formula has numerous real-world applications, including:
- Physics and engineering: The quadratic formula is used to model the motion of objects under the influence of gravity, friction, and other forces.
- Computer science: The quadratic formula is used in algorithms for solving systems of linear equations and for finding the roots of polynomials.
- Economics: The quadratic formula is used to model the behavior of economic systems, including the supply and demand curves.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It provides a systematic approach to finding the solutions to a quadratic equation, and it can be applied to a wide range of problems. By following the steps outlined in this article, you can apply the quadratic formula to a given equation and find the solutions.
Final Answer
The final answer is:
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Frequently Asked Questions
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 given by:
x = (-b ± √(b^2 - 4ac)) / 2a
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 in the given equation. Then, you can plug these values into the formula and simplify the expression to find the solutions.
Q: What are the common mistakes to avoid when applying the quadratic formula?
A: Some common mistakes to avoid when applying the quadratic formula include:
- Incorrectly identifying the values of a, b, and c: Make sure to carefully identify the values of a, b, and c in the given equation.
- Not simplifying the expression: Make sure to simplify the expression inside the square root to avoid unnecessary calculations.
- Not considering both the positive and negative cases: Make sure to consider both the positive and negative cases when solving for x.
Q: What are the real-world applications of the quadratic formula?
A: The quadratic formula has numerous real-world applications, including:
- Physics and engineering: The quadratic formula is used to model the motion of objects under the influence of gravity, friction, and other forces.
- Computer science: The quadratic formula is used in algorithms for solving systems of linear equations and for finding the roots of polynomials.
- Economics: The quadratic formula is used to model the behavior of economic systems, including the supply and demand curves.
Q: How do I choose between the positive and negative solutions?
A: When solving a quadratic equation, you may get two solutions: a positive solution and a negative solution. To choose between the two solutions, you need to consider the context of the problem. For example, if you are modeling the motion of an object, the negative solution may represent the object moving in the opposite direction.
Q: Can I use the quadratic formula to solve cubic equations?
A: No, the quadratic formula is only used to solve quadratic equations of the form ax^2 + bx + c = 0. If you have a cubic equation, you will need to use a different method to solve it.
Q: How do I simplify the expression inside the square root?
A: To simplify the expression inside the square root, you need to follow the order of operations (PEMDAS):
- Evaluate the expressions inside the parentheses.
- Evaluate any exponential expressions.
- Multiply and divide from left to right.
- Add and subtract from left to right.
Q: What is the difference between the quadratic formula and the factoring method?
A: The quadratic formula and the factoring method are two different methods for solving quadratic equations. The quadratic formula is a general method that can be used to solve any quadratic equation, while the factoring method is a specific method that can be used to solve quadratic equations that can be factored.
Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?
A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, you will need to use complex numbers to represent the solutions.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By following the steps outlined in this article, you can apply the quadratic formula to a given equation and find the solutions. Remember to avoid common mistakes and consider the real-world applications of the quadratic formula.
Final Answer
The final answer is: