Select The Correct Answer.Consider This Quadratic Equation:$\[ X^2 + 3 = 4x \\]Which Expression Correctly Sets Up The Quadratic Formula To Solve The Equation?A. \[$\frac{-(-4) \pm \sqrt{(-4)^2-4(1)(3)}}{2(1)}\$\]B. \[$\frac{-(-4)
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the quadratic formula and how to use it to solve quadratic equations. We will also examine the correct expression to set up the quadratic formula for a given equation.
What is the Quadratic Formula?
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = {\frac{-b \pm \sqrt{b^2-4ac}}{2a}$}$
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 x^2 + 3 = 4x, we can rewrite it as x^2 - 4x + 3 = 0. Comparing this equation to the standard form ax^2 + bx + c = 0, we can see that a = 1, b = -4, and c = 3.
Setting Up the Quadratic Formula
Now that we have identified the values of a, b, and c, we can set up the quadratic formula. Plugging in the values, we get:
x = {\frac{-(-4) \pm \sqrt{(-4)^2-4(1)(3)}}{2(1)}$}$
Evaluating the Expression
Let's evaluate the expression inside the square root:
(-4)^2 - 4(1)(3) = 16 - 12 = 4
Now, we can plug this value back into the expression:
x = {\frac{-(-4) \pm \sqrt{4}}{2(1)}$}$
x = {\frac{-(-4) \pm 2}{2}$}$
x = {\frac{4 \pm 2}{2}$}$
Simplifying the Expression
We can simplify the expression by evaluating the two possible values of x:
x = {\frac{4 + 2}{2}$ = 3 x = [$\frac{4 - 2}{2}$ = 1
Conclusion
In this article, we have explored the quadratic formula and how to use it to solve quadratic equations. We have also examined the correct expression to set up the quadratic formula for a given equation. By following the steps outlined in this article, students can master the quadratic formula and solve quadratic equations with ease.
Discussion
- What are some common mistakes students make when using the quadratic formula?
- How can students use the quadratic formula to solve real-world problems?
- What are some alternative methods for solving quadratic equations?
Answer Key
A. [\frac{-(-4) \pm \sqrt{(-4)^2-4(1)(3)}}{2(1)}\$}
This is the correct expression to set up the quadratic formula for the given equation.
Final Thoughts
Introduction
The quadratic formula is a powerful tool for solving quadratic equations, but it can be intimidating for students who are new to it. In this article, we will answer some of the most frequently asked questions about the quadratic formula, providing a deeper understanding of this mathematical 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 = {\frac{-b \pm \sqrt{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, plug these values into the formula and simplify the expression.
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. In the equation x^2 + 3 = 4x, we can rewrite it as x^2 - 4x + 3 = 0. Comparing this equation to the standard form ax^2 + bx + c = 0, we can see that a = 1, b = -4, and c = 3.
Q: How do I simplify the expression inside the square root?
A: To simplify the expression inside the square root, you need to evaluate the expression b^2 - 4ac. In the equation x^2 - 4x + 3 = 0, we have b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4.
Q: What are the two possible values of x?
A: The two possible values of x are given by:
x = {\frac{-b \pm \sqrt{b^2-4ac}}{2a}$}$
In the equation x^2 - 4x + 3 = 0, we have:
x = {\frac{-(-4) \pm \sqrt{4}}{2(1)}$}$
x = {\frac{4 \pm 2}{2}$}$
x = 3 or x = 1
Q: Can I use the quadratic formula to solve real-world problems?
A: Yes, the quadratic formula can be used to solve a wide range of real-world problems. For example, it can be used to model the trajectory of a projectile, the motion of a pendulum, or the growth of a population.
Q: What are some common mistakes students make when using the quadratic formula?
A: Some common mistakes students make when using the quadratic formula include:
- Not identifying the values of a, b, and c correctly
- Not simplifying the expression inside the square root correctly
- Not evaluating the two possible values of x correctly
Q: How can I practice using the quadratic formula?
A: You can practice using the quadratic formula by working through a series of examples and exercises. You can also use online resources, such as quadratic formula calculators or interactive quizzes, to help you practice.
Conclusion
The quadratic formula is a powerful tool for solving quadratic equations, but it can be intimidating for students who are new to it. By understanding the quadratic formula and how to use it, students can solve a wide range of problems and develop a deeper understanding of mathematical concepts.