What Is The First Step When Rewriting $y = -4x^2 + 2x - 7$ In The Form $y = A(x - H)^2 + K$?A. 2 Must Be Factored From $2x - 7$.B. -4 Must Be Factored From $-4x^2 + 2x$.C. $x$ Must Be Factored From

What is the First Step When Rewriting $y = -4x^2 + 2x - 7$ in the Form $y = a(x - h)^2 + k$?

Understanding the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is a powerful tool for analyzing and graphing quadratic functions. It is represented as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this form, the vertex is the minimum or maximum point of the parabola, depending on the sign of $a$. To rewrite a quadratic equation in vertex form, we need to complete the square, which involves manipulating the equation to create a perfect square trinomial.

The First Step: Factoring Out the Coefficient of $x^2$

When rewriting a quadratic equation in vertex form, the first step is to factor out the coefficient of $x^2$. This is because the coefficient of $x^2$ will become the value of $a$ in the vertex form. In the given equation $y = -4x^2 + 2x - 7$, the coefficient of $x^2$ is $-4$. To factor it out, we can write the equation as:

$$y = -4(x^2 - \frac{1}{2}x) - 7$$

Why Factoring Out the Coefficient of $x^2$ is the First Step

Factoring out the coefficient of $x^2$ is the first step because it allows us to create a perfect square trinomial inside the parentheses. This is a crucial step in completing the square, as it enables us to find the value of $h$ in the vertex form. By factoring out the coefficient of $x^2$, we are essentially isolating the quadratic term and preparing it for the next step in the process.

Conclusion

In conclusion, the first step when rewriting a quadratic equation in the form $y = a(x - h)^2 + k$ is to factor out the coefficient of $x^2$. This is a critical step in completing the square and finding the vertex of the parabola. By factoring out the coefficient of $x^2$, we can create a perfect square trinomial inside the parentheses and prepare it for the next step in the process.

Answer to the Question

Based on the above discussion, the correct answer to the question is:

B. -4 must be factored from $-4x^2 + 2x$.

This is because factoring out the coefficient of $x^2$ is the first step in rewriting a quadratic equation in vertex form.
Q&A: Completing the Square and Vertex Form of a Quadratic Equation

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is a powerful tool for analyzing and graphing quadratic functions. It is represented as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: Why is completing the square important in rewriting a quadratic equation in vertex form?

A: Completing the square is important because it allows us to create a perfect square trinomial inside the parentheses, which is a crucial step in finding the value of $h$ in the vertex form.

Q: What is the first step in completing the square?

A: The first step in completing the square is to factor out the coefficient of $x^2$. This is because the coefficient of $x^2$ will become the value of $a$ in the vertex form.

Q: How do I factor out the coefficient of $x^2$?

A: To factor out the coefficient of $x^2$, you can write the equation as $y = a(x^2 + bx + c) + d$, where $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term.

Q: What is the next step after factoring out the coefficient of $x^2$?

A: After factoring out the coefficient of $x^2$, the next step is to add and subtract the square of half the coefficient of $x$ inside the parentheses. This will create a perfect square trinomial.

Q: How do I add and subtract the square of half the coefficient of $x$?

A: To add and subtract the square of half the coefficient of $x$, you can write the equation as $y = a(x^2 + bx + \frac{b^2}{4} - \frac{b^2}{4}) + d$. Then, you can simplify the equation by combining like terms.

Q: What is the final step in completing the square?

A: The final step in completing the square is to simplify the equation and write it in vertex form. This will give you the value of $h$ and $k$ in the vertex form.

Q: How do I simplify the equation and write it in vertex form?

A: To simplify the equation and write it in vertex form, you can combine like terms and rewrite the equation as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not factoring out the coefficient of $x^2$
• Not adding and subtracting the square of half the coefficient of $x$
• Not simplifying the equation and writing it in vertex form
• Not checking the sign of $a$ to determine the direction of the parabola

Q: How can I practice completing the square and vertex form?

A: You can practice completing the square and vertex form by working through examples and exercises. You can also use online resources and calculators to help you visualize the process and check your work.

Q: What are some real-world applications of completing the square and vertex form?

A: Completing the square and vertex form have many real-world applications, including:

• Modeling population growth and decline
• Analyzing the motion of objects under the influence of gravity
• Designing and optimizing systems, such as electrical circuits and mechanical systems
• Solving problems in physics, engineering, and economics