Find The Values Of \[$a\$\], \[$b\$\], And \[$c\$\] That Should Be Used In The Quadratic Formula To Determine The Solution Of The Quadratic Equation \[$-x^2 + 7x + 4 = 0\$\].\[$a = \square\$\] \[$b =
Introduction
The quadratic formula is a powerful tool used to solve quadratic equations of the form ax^2 + bx + c = 0. It is a fundamental concept in algebra and is widely used in various fields such as physics, engineering, and economics. In this article, we will focus on finding the values of a, b, and c that should be used in the quadratic formula to determine the solution of the quadratic equation -x^2 + 7x + 4 = 0.
Understanding the Quadratic Formula
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. To use this formula, we need to identify the values of a, b, and c from the given quadratic equation.
Identifying the Values of a, b, and c
The given quadratic equation is -x^2 + 7x + 4 = 0. To identify the values of a, b, and c, we need to compare this equation with the general form of a quadratic equation, which is ax^2 + bx + c = 0.
- a: The coefficient of x^2 is -1, so a = -1.
- b: The coefficient of x is 7, so b = 7.
- c: The constant term is 4, so c = 4.
Substituting the Values of a, b, and c into the Quadratic Formula
Now that we have identified the values of a, b, and c, we can substitute them into the quadratic formula:
x = (-(7) ± √((7)^2 - 4(-1)(4))) / 2(-1)
Simplifying the expression, we get:
x = (-7 ± √(49 + 16)) / -2
x = (-7 ± √65) / -2
Simplifying the Expression
To simplify the expression further, we can use the fact that √65 is an irrational number. However, we can simplify the expression by factoring out -2 from the numerator:
x = (-7 ± √65) / -2
x = (7 ∓ √65) / 2
Conclusion
In this article, we have found the values of a, b, and c that should be used in the quadratic formula to determine the solution of the quadratic equation -x^2 + 7x + 4 = 0. We have identified the values of a, b, and c by comparing the given equation with the general form of a quadratic equation. We have then substituted these values into the quadratic formula and simplified the expression to obtain the final solution.
Final Answer
The final answer is:
a = -1 b = 7 c = 4
Example Use Case
The quadratic formula is widely used in various fields such as physics, engineering, and economics. For example, in physics, the quadratic formula is used to solve problems involving motion under constant acceleration. In engineering, the quadratic formula is used to design and optimize systems such as bridges and buildings.
Tips and Tricks
- When using the quadratic formula, make sure to identify the values of a, b, and c correctly.
- Simplify the expression by factoring out common terms.
- Use the fact that √65 is an irrational number to simplify the expression further.
Common Mistakes
- Failing to identify the values of a, b, and c correctly.
- Not simplifying the expression by factoring out common terms.
- Not using the fact that √65 is an irrational number to simplify the expression further.
Conclusion
Introduction
The quadratic formula is a powerful tool used to solve quadratic equations of the form ax^2 + bx + c = 0. In our previous article, we discussed how to find the values of a, b, and c that should be used in the quadratic formula to determine the solution of the quadratic equation -x^2 + 7x + 4 = 0. In this article, we will answer some frequently asked questions about the quadratic formula.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax^2 + bx + c = 0. It is given by:
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 from the given quadratic equation. Then, substitute these values into the quadratic formula and simplify the expression to obtain the final solution.
Q: What are the values of a, b, and c in the quadratic formula?
A: The values of a, b, and c are the coefficients of the quadratic equation. For example, in the quadratic equation -x^2 + 7x + 4 = 0, a = -1, b = 7, and c = 4.
Q: How do I simplify the expression in the quadratic formula?
A: To simplify the expression, you can factor out common terms and use the fact that √65 is an irrational number.
Q: What is the final answer to the quadratic equation -x^2 + 7x + 4 = 0?
A: The final answer is:
x = (7 ∓ √65) / 2
Q: Can I use the quadratic formula to solve any quadratic equation?
A: Yes, you can use the quadratic formula to solve any quadratic equation of the form ax^2 + bx + c = 0.
Q: What are some common mistakes to avoid when using the quadratic formula?
A: Some common mistakes to avoid when using the quadratic formula include:
- Failing to identify the values of a, b, and c correctly.
- Not simplifying the expression by factoring out common terms.
- Not using the fact that √65 is an irrational number to simplify the expression further.
Q: How do I apply the quadratic formula in real-life situations?
A: The quadratic formula is widely used in various fields such as physics, engineering, and economics. For example, in physics, the quadratic formula is used to solve problems involving motion under constant acceleration. In engineering, the quadratic formula is used to design and optimize systems such as bridges and buildings.
Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?
A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression and using the fact that √65 is an irrational number.
Conclusion
In conclusion, the quadratic formula is a powerful tool used to solve quadratic equations of the form ax^2 + bx + c = 0. By identifying the values of a, b, and c and substituting them into the quadratic formula, we can determine the solution of the quadratic equation. We have answered some frequently asked questions about the quadratic formula and provided some tips and tricks for using it effectively.
Final Answer
The final answer is:
a = -1 b = 7 c = 4
Example Use Case
The quadratic formula is widely used in various fields such as physics, engineering, and economics. For example, in physics, the quadratic formula is used to solve problems involving motion under constant acceleration. In engineering, the quadratic formula is used to design and optimize systems such as bridges and buildings.
Tips and Tricks
- When using the quadratic formula, make sure to identify the values of a, b, and c correctly.
- Simplify the expression by factoring out common terms.
- Use the fact that √65 is an irrational number to simplify the expression further.
Common Mistakes
- Failing to identify the values of a, b, and c correctly.
- Not simplifying the expression by factoring out common terms.
- Not using the fact that √65 is an irrational number to simplify the expression further.