Factoring out the wrong coefficient

Oct 13, 2025 by miodeya.com 78 views

Introduction

Completing the square is a powerful technique used to rewrite quadratic equations in the form $y = a(x-h)^2 + k$ . This form is particularly useful for solving quadratic equations, as it allows us to easily identify the vertex of the parabola represented by the equation. In this article, we will explore the first step in completing the square for the quadratic equation $y = 3x^2 + 9x - 18$ .

Understanding the Basics

Before we dive into the first step, let's review the basics of completing the square. The process involves rewriting a quadratic equation in the form $y = a(x-h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The key to completing the square is to create a perfect square trinomial on the left-hand side of the equation.

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

The first step in completing the square is to factor out the coefficient of $x^2$ from the quadratic term. In the equation $y = 3x^2 + 9x - 18$ , the coefficient of $x^2$ is 3. To factor this out, we can write the equation as:

y = 3(x^2 + 3x) - 18

This step is crucial, as it allows us to create a perfect square trinomial on the left-hand side of the equation.

Why Factoring Out the Coefficient of $x^2$ is Important

Factoring out the coefficient of $x^2$ is important because it allows us to create a perfect square trinomial on the left-hand side of the equation. A perfect square trinomial is a trinomial that can be written in the form $(x + b)^2$ , where $b$ is a constant. By factoring out the coefficient of $x^2$ , we can create a perfect square trinomial that will help us complete the square.

Common Mistakes to Avoid

When factoring out the coefficient of $x^2$ , it's easy to make mistakes. Here are a few common mistakes to avoid:

Not factoring out the coefficient of $x^2$ : If you don't factor out the coefficient of $x^2$ , you won't be able to create a perfect square trinomial on the left-hand side of the equation.
Factoring out the wrong coefficient: Make sure you factor out the correct coefficient of $x^2$ . In this case, the coefficient of $x^2$ is 3, not 9 or 18.

Conclusion

In conclusion, the first step in completing the square for the quadratic equation $y = 3x^2 + 9x - 18$ is to factor out the coefficient of $x^2$ . This step is crucial, as it allows us to create a perfect square trinomial on the left-hand side of the equation. By following this step, we can complete the square and rewrite the equation in the form $y = a(x-h)^2 + k$ .

Additional Tips and Resources

If you're struggling to complete the square, here are a few additional tips and resources that may help:

Practice, practice, practice: The more you practice completing the square, the more comfortable you'll become with the process.
Use online resources: There are many online resources available that can help you learn how to complete the square, including video tutorials and practice problems.
Seek help from a tutor: If you're really struggling to complete the square, consider seeking help from a tutor. A tutor can provide you with one-on-one instruction and help you work through difficult problems.

Final Thoughts

Introduction

Completing the square is a powerful technique used to rewrite quadratic equations in the form $y = a(x-h)^2 + k$ . In our previous article, we explored the first step in completing the square for the quadratic equation $y = 3x^2 + 9x - 18$ . In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to rewrite a quadratic equation in the form $y = a(x-h)^2 + k$ . This form is particularly useful for solving quadratic equations, as it allows us to easily identify the vertex of the parabola represented by the equation.

Q: How do I know when to complete the square?

A: You should complete the square whenever you need to rewrite a quadratic equation in the form $y = a(x-h)^2 + k$ . This is often the case when you need to solve a quadratic equation or find the vertex of a parabola.

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 $x^2$ from the quadratic term. This will create a perfect square trinomial on the left-hand side of the equation.

Q: Why do I need to factor out the coefficient of $x^2$ ?

A: You need to factor out the coefficient of $x^2$ because it allows you to create a perfect square trinomial on the left-hand side of the equation. A perfect square trinomial is a trinomial that can be written in the form $(x + b)^2$ , where $b$ is a constant.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be written in the form $(x + b)^2$ , where $b$ is a constant. It is called "perfect" because it can be factored into a squared binomial.

Q: How do I create a perfect square trinomial?

A: To create a perfect square trinomial, you need to factor out the coefficient of $x^2$ from the quadratic term and then add and subtract the square of half the coefficient of $x$ .

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

A: The next step in completing the square is to add and subtract the square of half the coefficient of $x$ to the equation. This will create a perfect square trinomial on the left-hand side of the equation.

Q: Why do I need to add and subtract the square of half the coefficient of $x$ ?

A: You need to add and subtract the square of half the coefficient of $x$ because it allows you to create a perfect square trinomial on the left-hand side of the equation. The square of half the coefficient of $x$ is equal to the constant term in the perfect square trinomial.

Q: How do I find the value of $h$ in the equation $y = a(x-h)^2 + k$ ?

A: To find the value of $h$ in the equation $y = a(x-h)^2 + k$ , you need to look at the coefficient of $x$ in the perfect square trinomial. The value of $h$ is equal to the negative of half the coefficient of $x$ .

Q: How do I find the value of $k$ in the equation $y = a(x-h)^2 + k$ ?

A: To find the value of $k$ in the equation $y = a(x-h)^2 + k$ , you need to look at the constant term in the perfect square trinomial. The value of $k$ is equal to the constant term minus the square of half the coefficient of $x$ .

Conclusion

In conclusion, completing the square is a powerful technique used to rewrite quadratic equations in the form $y = a(x-h)^2 + k$ . By following the steps outlined in this article, you can complete the square and rewrite the equation in this form. Remember to factor out the coefficient of $x^2$ , create a perfect square trinomial, and add and subtract the square of half the coefficient of $x$ . With practice and patience, you'll become a pro at completing the square in no time!