Solve $0=4x^2+12x+9$.Select The Equation That Shows The Correct Substitution Of $a, B$, And \$c$[/tex\] In The Quadratic Formula:A. $x=\frac{12 \pm \sqrt{12^2-4(4)(9)}}{2(4)}$B. $x=\frac{-12 \pm
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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 focus on solving a quadratic equation using the quadratic formula and explore the correct substitution of coefficients in the 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
Solving the Given Quadratic Equation
The given quadratic equation is:
0 = 4x^2 + 12x + 9
To solve this equation, we can use the quadratic formula. First, we need to identify the coefficients a, b, and c in the equation.
- a = 4 (coefficient of x^2)
- b = 12 (coefficient of x)
- c = 9 (constant term)
Substituting Coefficients in the Quadratic Formula
Now that we have identified the coefficients, we can substitute them into the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
x = (-(12) ± √((12)^2 - 4(4)(9))) / 2(4)
Simplifying the expression, we get:
x = (-12 ± √(144 - 144)) / 8
x = (-12 ± √0) / 8
x = (-12 ± 0) / 8
x = -12 / 8
x = -3/2
Conclusion
In this article, we solved a quadratic equation using the quadratic formula and explored the correct substitution of coefficients in the formula. We identified the coefficients a, b, and c in the given equation and substituted them into the quadratic formula. The solutions to the equation were found to be x = -3/2.
Discussion
The quadratic formula is a powerful tool for solving quadratic equations. However, it requires careful substitution of coefficients to obtain the correct solutions. In this case, we substituted the coefficients a, b, and c into the quadratic formula and obtained the correct solutions.
Final Answer
The final answer is:
x = -3/2
This is the correct solution to the given quadratic equation.
Step-by-Step Solution
Here is the step-by-step solution to the problem:
- Identify the coefficients a, b, and c in the given equation.
- Substitute the values of a, b, and c into the quadratic formula.
- Simplify the expression to obtain the solutions to the equation.
Common Mistakes
When solving quadratic equations using the quadratic formula, it is common to make mistakes in substituting coefficients or simplifying the expression. To avoid these mistakes, it is essential to carefully read and follow the instructions.
Tips and Tricks
Here are some tips and tricks for solving quadratic equations using the quadratic formula:
- Make sure to identify the coefficients a, b, and c correctly.
- Substitute the values of a, b, and c into the quadratic formula carefully.
- Simplify the expression step by step to avoid mistakes.
Real-World Applications
Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, quadratic equations are used to model the motion of objects under the influence of gravity, to design electrical circuits, and to analyze economic data.
Conclusion
In conclusion, solving quadratic equations using the quadratic formula requires careful substitution of coefficients and simplification of the expression. By following the step-by-step solution and avoiding common mistakes, we can obtain the correct solutions to quadratic equations. The quadratic formula is a powerful tool for solving quadratic equations, and its applications are numerous in real-world scenarios.
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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 provide a Q&A guide to help you understand quadratic equations and their solutions.
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. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
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 coefficients a, b, and c in the given equation. Then, substitute these values into the quadratic formula and simplify the expression to obtain the solutions.
Q: What are the steps to solve a quadratic equation?
A: Here are the steps to solve a quadratic equation:
- Identify the coefficients a, b, and c in the given equation.
- Substitute the values of a, b, and c into the quadratic formula.
- Simplify the expression to obtain the solutions.
Q: What is the difference between the two solutions obtained from the quadratic formula?
A: The two solutions obtained from the quadratic formula are called the roots of the equation. They are denoted by x = (-b + √(b^2 - 4ac)) / 2a and x = (-b - √(b^2 - 4ac)) / 2a. These roots are the values of x that satisfy the equation.
Q: Can a quadratic equation have more than two solutions?
A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula provides two roots, and these roots are the only solutions to the equation.
Q: What is the significance of the discriminant in the quadratic formula?
A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: Can a quadratic equation have no real solutions?
A: Yes, a quadratic equation can have no real solutions. This occurs when the discriminant is negative, which means the expression b^2 - 4ac is less than zero.
Q: What is the relationship between the quadratic formula and the factoring method?
A: The quadratic formula and the factoring method are two different ways of solving quadratic equations. The factoring method involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.
Q: Can the quadratic formula be used to solve all types of quadratic equations?
A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including those that cannot be factored.
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:
- Not identifying the coefficients a, b, and c correctly.
- Not substituting the values of a, b, and c into the quadratic formula correctly.
- Not simplifying the expression correctly.
Q: How can I practice solving quadratic equations using the quadratic formula?
A: You can practice solving quadratic equations using the quadratic formula by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the quadratic formula and its applications, you can solve a wide range of problems in mathematics and other fields. Remember to identify the coefficients a, b, and c correctly, substitute them into the quadratic formula, and simplify the expression to obtain the solutions. With practice and patience, you can become proficient in using the quadratic formula to solve quadratic equations.