Rewrite $5x^2 - 20x + 6$ In Vertex Form.

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Introduction

In algebra, the vertex form of a quadratic function is a way to express the function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for identifying the vertex and the direction of the parabola. In this article, we will learn how to rewrite the quadratic function $5x^2 - 20x + 6$ in vertex form.

Understanding the Standard Form

Before we can rewrite the quadratic function in vertex form, we need to understand the standard form of a quadratic function. The standard form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the quadratic function is $5x^2 - 20x + 6$, where $a = 5$, $b = -20$, and $c = 6$.

Completing the Square

To rewrite the quadratic function in vertex form, we need to complete the square. Completing the square involves creating a perfect square trinomial from the quadratic function. To do this, we need to find the value of $x$ that makes the expression $ax^2 + bx$ a perfect square trinomial.

The formula for completing the square is:

$$x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$$

In this case, we have:

$$5x^2 - 20x = 5\left(x^2 - 4x\right)$$

To complete the square, we need to add and subtract $\left(\frac{b}{2}\right)^2$ inside the parentheses:

$$5\left(x^2 - 4x + 4 - 4\right)$$

Now, we can factor the perfect square trinomial:

$$5\left(x - 2\right)^2 - 20$$

Adding the Constant Term

Now that we have completed the square, we can add the constant term $6$ to the expression:

$$5\left(x - 2\right)^2 - 20 + 6$$

Simplifying the expression, we get:

$$5\left(x - 2\right)^2 - 14$$

Rewriting in Vertex Form

Now that we have completed the square and added the constant term, we can rewrite the quadratic function in vertex form:

$$y = 5\left(x - 2\right)^2 - 14$$

This is the vertex form of the quadratic function $5x^2 - 20x + 6$. The vertex of the parabola is $(2, -14)$.

Conclusion

In this article, we learned how to rewrite the quadratic function $5x^2 - 20x + 6$ in vertex form. We used the method of completing the square to create a perfect square trinomial from the quadratic function. We then added the constant term to the expression and simplified it to get the vertex form of the quadratic function. The vertex form of a quadratic function is a useful way to express the function and identify the vertex and direction of the parabola.

Example Problems

Here are some example problems that you can try to practice rewriting quadratic functions in vertex form:

• Rewrite the quadratic function $x^2 + 6x + 8$ in vertex form.
• Rewrite the quadratic function $2x^2 - 8x + 3$ in vertex form.
• Rewrite the quadratic function $3x^2 + 12x - 2$ in vertex form.

Tips and Tricks

Here are some tips and tricks that you can use to help you rewrite quadratic functions in vertex form:

• Make sure to complete the square correctly by adding and subtracting the same value inside the parentheses.
• Make sure to add the constant term to the expression after completing the square.
• Simplify the expression carefully to get the vertex form of the quadratic function.

Common Mistakes

Here are some common mistakes that you can avoid when rewriting quadratic functions in vertex form:

• Not completing the square correctly by adding and subtracting the same value inside the parentheses.
• Not adding the constant term to the expression after completing the square.
• Not simplifying the expression carefully to get the vertex form of the quadratic function.

Real-World Applications

Rewriting quadratic functions in vertex form has many real-world applications. Here are a few examples:

• Optimization: Vertex form is useful for identifying the maximum or minimum value of a quadratic function, which is important in optimization problems.
• Physics: Vertex form is useful for modeling the motion of objects under the influence of gravity or other forces.
• Engineering: Vertex form is useful for designing and analyzing the performance of systems, such as bridges or buildings.

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about rewriting quadratic functions in vertex form:

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

A: The vertex form of a quadratic function is a way to express the function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I rewrite a quadratic function in vertex form?

A: To rewrite a quadratic function in vertex form, you need to complete the square and add the constant term. This involves creating a perfect square trinomial from the quadratic function and then adding the constant term to the expression.

Q: What is completing the square?

A: Completing the square is a process of creating a perfect square trinomial from a quadratic expression. This involves adding and subtracting the same value inside the parentheses to create a perfect square trinomial.

Q: How do I complete the square?

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

1. Identify the coefficient of the $x^2$ term.
2. Identify the coefficient of the $x$ term.
3. Add and subtract the square of half the coefficient of the $x$ term inside the parentheses.
4. Factor the perfect square trinomial.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.

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

A: To find the vertex of a parabola, you need to rewrite the quadratic function in vertex form. The vertex is the point $(h, k)$ in the vertex form of the quadratic function.

Q: What are some real-world applications of rewriting quadratic functions in vertex form?

A: Rewriting quadratic functions in vertex form has many real-world applications, including:

• Optimization: Vertex form is useful for identifying the maximum or minimum value of a quadratic function, which is important in optimization problems.
• Physics: Vertex form is useful for modeling the motion of objects under the influence of gravity or other forces.
• Engineering: Vertex form is useful for designing and analyzing the performance of systems, such as bridges or buildings.

Q: What are some common mistakes to avoid when rewriting quadratic functions in vertex form?

A: Some common mistakes to avoid when rewriting quadratic functions in vertex form include:

• Not completing the square correctly by adding and subtracting the same value inside the parentheses.
• Not adding the constant term to the expression after completing the square.
• Not simplifying the expression carefully to get the vertex form of the quadratic function.

Q: How can I practice rewriting quadratic functions in vertex form?

A: You can practice rewriting quadratic functions in vertex form by working through example problems and exercises. You can also use online resources and tools to help you practice and learn.

Q: What are some tips and tricks for rewriting quadratic functions in vertex form?

A: Some tips and tricks for rewriting quadratic functions in vertex form include:

• Make sure to complete the square correctly by adding and subtracting the same value inside the parentheses.
• Make sure to add the constant term to the expression after completing the square.
• Simplify the expression carefully to get the vertex form of the quadratic function.

Conclusion

In conclusion, rewriting quadratic functions in vertex form is an important skill that has many real-world applications. By completing the square and adding the constant term, we can rewrite a quadratic function in vertex form and identify the vertex and direction of the parabola. With practice and patience, you can master this skill and apply it to a wide range of problems.