Complete The Square To Re-write The Quadratic Function In Vertex Form.$y = X^2 + 9x - 6$

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in vertex form is crucial for solving various mathematical problems. The vertex form of a quadratic function is a powerful tool for analyzing the behavior of the function, including its maximum or minimum value, and the direction of its opening. In this article, we will explore how to complete the square to rewrite the quadratic function $y = x^2 + 9x - 6$ in vertex form.

What is Completing the Square?

Completing the square is a mathematical technique used to rewrite a quadratic function in a specific form, known as the vertex form. This technique involves manipulating the quadratic function to create a perfect square trinomial, which can be factored into the square of a binomial. The vertex form of a quadratic function is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Step 1: Identify the Coefficients

To complete the square, we need to identify the coefficients of the quadratic function. In the given function $y = x^2 + 9x - 6$, the coefficient of $x^2$ is 1, the coefficient of $x$ is 9, and the constant term is -6.

Step 2: Move the Constant Term

The next step is to move the constant term to the right-hand side of the equation. This will give us:

$$y + 6 = x^2 + 9x$$

Step 3: Find the Value to Add

To complete the square, we need to find the value to add to both sides of the equation. This value is given by:

$$\left(\frac{b}{2}\right)^2 = \left(\frac{9}{2}\right)^2 = \frac{81}{4}$$

where $b$ is the coefficient of $x$.

Step 4: Add the Value to Both Sides

Now, we add the value we found in the previous step to both sides of the equation:

$$y + 6 + \frac{81}{4} = x^2 + 9x + \frac{81}{4}$$

Step 5: Factor the Perfect Square Trinomial

The left-hand side of the equation is now a perfect square trinomial, which can be factored into the square of a binomial:

$$y + 6 + \frac{81}{4} = \left(x + \frac{9}{2}\right)^2$$

Step 6: Simplify the Equation

Finally, we simplify the equation by subtracting the constant term from both sides:

$$y = \left(x + \frac{9}{2}\right)^2 - 6 - \frac{81}{4}$$

Simplifying the Equation Further

We can simplify the equation further by combining the constant terms:

$$y = \left(x + \frac{9}{2}\right)^2 - \frac{24}{4}$$

$$y = \left(x + \frac{9}{2}\right)^2 - 6$$

The Final Answer

Therefore, the quadratic function $y = x^2 + 9x - 6$ can be rewritten in vertex form as:

$$y = \left(x + \frac{9}{2}\right)^2 - 6$$

Conclusion

In this article, we have learned how to complete the square to rewrite the quadratic function $y = x^2 + 9x - 6$ in vertex form. We have identified the coefficients of the quadratic function, moved the constant term to the right-hand side, found the value to add to both sides, added the value to both sides, factored the perfect square trinomial, and simplified the equation. The final answer is the vertex form of the quadratic function, which is given by:

$$y = \left(x + \frac{9}{2}\right)^2 - 6$$

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is a powerful tool for analyzing the behavior of the function. It provides information about the vertex of the parabola, which is the maximum or minimum point of the function. The vertex form of a quadratic function is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Example Problems

Here are some example problems that demonstrate how to complete the square to rewrite quadratic functions in vertex form:

• $y = x^2 + 4x - 5$
• $y = x^2 - 2x - 3$
• $y = x^2 + 6x - 2$

Tips and Tricks

Here are some tips and tricks for completing the square:

• Make sure to identify the coefficients of the quadratic function correctly.
• Move the constant term to the right-hand side of the equation.
• Find the value to add to both sides of the equation.
• Add the value to both sides of the equation.
• Factor the perfect square trinomial.
• Simplify the equation.

Conclusion

Introduction

Completing the square is a mathematical technique used to rewrite quadratic functions in vertex form. In our previous article, we explored how to complete the square to rewrite the quadratic function $y = x^2 + 9x - 6$ in vertex form. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite quadratic functions in vertex form. It involves manipulating the quadratic function to create a perfect square trinomial, which can be factored into the square of a binomial.

Q: Why is completing the square important?

A: Completing the square is important because it provides information about the vertex of the parabola, which is the maximum or minimum point of the function. This information is crucial for analyzing the behavior of the function.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

1. Identify the coefficients of the quadratic function.
2. Move the constant term to the right-hand side of the equation.
3. Find the value to add to both sides of the equation.
4. Add the value to both sides of the equation.
5. Factor the perfect square trinomial.
6. Simplify the equation.

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

A: The vertex form of a quadratic function is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to complete the square and rewrite the quadratic function in vertex form. The vertex of the parabola is given by the values of $h$ and $k$ in the vertex form of the quadratic function.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different techniques used to rewrite quadratic functions. Factoring involves expressing the quadratic function as a product of two binomials, while completing the square involves manipulating the quadratic function to create a perfect square trinomial.

Q: Can I use completing the square to solve quadratic equations?

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the quadratic equation in vertex form, you can find the values of $x$ that satisfy the equation.

Q: Are there any tips and tricks for completing the square?

A: Yes, here are some tips and tricks for completing the square:

• Make sure to identify the coefficients of the quadratic function correctly.
• Move the constant term to the right-hand side of the equation.
• Find the value to add to both sides of the equation.
• Add the value to both sides of the equation.
• Factor the perfect square trinomial.
• Simplify the equation.

Q: Can I use completing the square to graph quadratic functions?

A: Yes, you can use completing the square to graph quadratic functions. By rewriting the quadratic function in vertex form, you can find the vertex of the parabola and use it to graph the function.

Conclusion

In conclusion, completing the square is a powerful technique used to rewrite quadratic functions in vertex form. It provides information about the vertex of the parabola, which is the maximum or minimum point of the function. By following the steps outlined in this article, you can complete the square and answer some frequently asked questions about this technique.

Example Problems

Here are some example problems that demonstrate how to complete the square to rewrite quadratic functions in vertex form:

• $y = x^2 + 4x - 5$
• $y = x^2 - 2x - 3$
• $y = x^2 + 6x - 2$

Practice Problems

Here are some practice problems that you can use to practice completing the square:

• $y = x^2 + 3x - 2$
• $y = x^2 - 5x + 6$
• $y = x^2 + 2x - 1$

Answer Key

Here are the answers to the practice problems:

• $y = (x + 1)^2 - 3$
• $y = (x - 3)^2 - 4$
• $y = (x + 1)^2 - 2$