Select The Correct Answer.Consider The Quadratic Equation Below.$\[4x^2 - 5 = 3x + 4\\]Determine The Correct Setup For Solving The Equation Using The Quadratic Formula.A. $\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(9)}}{2(4)}\\]B.
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 the quadratic formula and provide a step-by-step guide on how to solve quadratic equations using this formula. We will also discuss the correct setup for solving the equation using the quadratic formula.
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
Understanding the Quadratic Formula
To use the quadratic formula, we need to identify the values of a, b, and c in the quadratic equation. In the equation 4x^2 - 5 = 3x + 4, we can rewrite it in the standard form as 4x^2 - 3x - 9 = 0. Now, we can identify the values of a, b, and c as follows:
a = 4 b = -3 c = -9
Setting Up the Quadratic Formula
Now that we have identified the values of a, b, and c, we can set up the quadratic formula. We will use the formula x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in the values of a, b, and c, we get:
x = (-( -3) ± √((-3)^2 - 4(4)(-9))) / 2(4)
Simplifying the Expression
To simplify the expression, we need to follow the order of operations (PEMDAS):
- Evaluate the expressions inside the parentheses: (-3)^2 = 9 and 4(4)(-9) = -144
- Simplify the expression inside the square root: 9 - (-144) = 153
- Simplify the expression inside the parentheses: -( -3) = 3
- Simplify the expression inside the square root: √153
The Correct Setup
The correct setup for solving the equation using the quadratic formula is:
x = (3 ± √153) / 8
Conclusion
In this article, we have discussed the quadratic formula and provided a step-by-step guide on how to solve quadratic equations using this formula. We have also discussed the correct setup for solving the equation using the quadratic formula. By following the steps outlined in this article, students and professionals can confidently solve quadratic equations and apply the quadratic formula to a wide range of mathematical problems.
Common Mistakes to Avoid
When setting up the quadratic formula, it is easy to make mistakes. Here are some common mistakes to avoid:
- Incorrectly identifying the values of a, b, and c: Make sure to identify the values of a, b, and c correctly in the quadratic equation.
- Incorrectly simplifying the expression: Make sure to follow the order of operations (PEMDAS) when simplifying the expression.
- Incorrectly setting up the quadratic formula: Make sure to set up the quadratic formula correctly using the values of a, b, and c.
Real-World Applications
The quadratic formula has numerous real-world applications in fields such as physics, engineering, and economics. Some examples include:
- Projectile motion: The quadratic formula can be used to model the trajectory of a projectile under the influence of gravity.
- Optimization problems: The quadratic formula can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
- Economic modeling: The quadratic formula can be used to model economic systems, such as the supply and demand curve.
Conclusion
Introduction
The quadratic formula is a fundamental concept in mathematics, and it can be a bit tricky to understand and apply. In this article, we will answer some of the most frequently asked questions about the quadratic formula, providing a clear and concise explanation of each concept.
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 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 can plug these values into the formula and simplify the expression to find the solutions.
Q: What are the values of a, b, and c?
A: The values of a, b, and c are the coefficients of the quadratic equation. In the equation ax^2 + bx + c = 0, a is the coefficient of x^2, b is the coefficient of x, and c is the constant term.
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). First, evaluate the expressions inside the parentheses, then simplify the expression inside the square root.
Q: What is the difference between the two solutions?
A: The two solutions of the quadratic equation are given by:
x = (-b + √(b^2 - 4ac)) / 2a x = (-b - √(b^2 - 4ac)) / 2a
The difference between the two solutions is the sign of the square root term. If the square root term is positive, the solutions are real and distinct. If the square root term is negative, the solutions are complex and conjugate.
Q: What is the discriminant?
A: The discriminant is the expression inside the square root of the quadratic formula:
b^2 - 4ac
The discriminant determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex and conjugate.
Q: How do I determine the nature of the solutions?
A: To determine the nature of the solutions, you need to evaluate the discriminant. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex and conjugate.
Q: What are the real-world applications of the quadratic formula?
A: The quadratic formula has numerous real-world applications in fields such as physics, engineering, and economics. Some examples include:
- Projectile motion: The quadratic formula can be used to model the trajectory of a projectile under the influence of gravity.
- Optimization problems: The quadratic formula can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
- Economic modeling: The quadratic formula can be used to model economic systems, such as the supply and demand curve.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the concepts and techniques outlined in this article, you can confidently apply the quadratic formula to a wide range of mathematical problems. Remember to practice and review the material to become proficient in using the quadratic formula.