Complete The Square To Find The Vertex Of This Parabola.$ X^2 - 4y - 12x + 68 = 0 }$Vertex { (?, ?)$ $

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Introduction

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In mathematics, a parabola is a quadratic equation that can be represented in the form of $y = ax^2 + bx + c$. The vertex of a parabola is the point at which the parabola changes direction, and it is a crucial concept in algebra and calculus. In this article, we will discuss how to find the vertex of a parabola using the method of completing the square.

What is Completing the Square?

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Completing the square is a mathematical technique used to rewrite a quadratic equation in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into the form of $(x - h)^2$.

The Formula for Completing the Square

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The formula for completing the square is as follows:

$$y = ax^2 + bx + c \Rightarrow y = a(x - h)^2 + k$$

where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

Step-by-Step Guide to Finding the Vertex

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To find the vertex of a parabola using the method of completing the square, follow these steps:

Step 1: Write the Equation in Standard Form

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The given equation is $x^2 - 4y - 12x + 68 = 0$. To write this equation in standard form, we need to rearrange the terms to get:

$$x^2 - 12x - 4y + 68 = 0$$

Step 2: Group the Terms

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Group the terms that contain $x$ and the constant term:

$$x^2 - 12x + 68 = 4y$$

Step 3: Move the Constant Term to the Right Side

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Move the constant term to the right side of the equation:

$$x^2 - 12x = 4y - 68$$

Step 4: Add and Subtract the Square of Half the Coefficient of $x$

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Add and subtract the square of half the coefficient of $x$ to the left side of the equation:

$$x^2 - 12x + 36 = 4y - 68 - 36$$

Step 5: Factor the Perfect Square Trinomial

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Factor the perfect square trinomial on the left side of the equation:

$$(x - 6)^2 = 4y - 104$$

Step 6: Divide Both Sides by 4

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Divide both sides of the equation by 4:

$$\frac{(x - 6)^2}{4} = y - 26$$

Step 7: Add 26 to Both Sides

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Add 26 to both sides of the equation:

$$\frac{(x - 6)^2}{4} + 26 = y$$

Step 8: Write the Equation in Vertex Form

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Write the equation in vertex form:

$$y = \frac{(x - 6)^2}{4} + 26$$

Finding the Vertex

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The vertex of the parabola is the point at which the parabola changes direction. In this case, the vertex is the point $(h, k)$, where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

In this case, $a = \frac{1}{4}$, $b = -6$, and $c = 26$. Plugging these values into the formulas, we get:

$$h = -\frac{-6}{2 \cdot \frac{1}{4}} = 12$$

$$k = 26 - \frac{(-6)^2}{4 \cdot \frac{1}{4}} = 26 - 36 = -10$$

Therefore, the vertex of the parabola is the point $(12, -10)$.

Conclusion

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In this article, we discussed how to find the vertex of a parabola using the method of completing the square. We started with the given equation $x^2 - 4y - 12x + 68 = 0$ and rearranged it to write it in standard form. We then grouped the terms, moved the constant term to the right side, added and subtracted the square of half the coefficient of $x$, factored the perfect square trinomial, divided both sides by 4, added 26 to both sides, and wrote the equation in vertex form. Finally, we found the vertex of the parabola by plugging the values of $a$, $b$, and $c$ into the formulas for $h$ and $k$. The vertex of the parabola is the point $(12, -10)$.

Example Problems

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Problem 1

Find the vertex of the parabola represented by the equation $y = x^2 + 4x + 3$.

Solution

To find the vertex of the parabola, we need to complete the square. First, we write the equation in standard form:

$$x^2 + 4x + 3 = 0$$

Next, we group the terms:

$$x^2 + 4x = -3$$

Then, we add and subtract the square of half the coefficient of $x$:

$$x^2 + 4x + 4 = -3 + 4$$

Now, we factor the perfect square trinomial:

$$(x + 2)^2 = 1$$

Finally, we divide both sides by 1 and add 1 to both sides:

$$(x + 2)^2 = 1 \Rightarrow y = (x + 2)^2 - 1$$

The vertex of the parabola is the point $(-2, -1)$.

Problem 2

Find the vertex of the parabola represented by the equation $y = 2x^2 - 6x + 1$.

Solution

To find the vertex of the parabola, we need to complete the square. First, we write the equation in standard form:

$$2x^2 - 6x + 1 = 0$$

Next, we group the terms:

$$2x^2 - 6x = -1$$

Then, we add and subtract the square of half the coefficient of $x$:

$$2x^2 - 6x + 9 = -1 + 9$$

Now, we factor the perfect square trinomial:

$$2(x - 3/2)^2 = 8$$

Finally, we divide both sides by 2 and add 4 to both sides:

$$2(x - 3/2)^2 = 8 \Rightarrow y = 2(x - 3/2)^2 + 4$$

The vertex of the parabola is the point $(\frac{3}{2}, 4)$.

Final Thoughts

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In conclusion, completing the square is a powerful technique for finding the vertex of a parabola. By following the steps outlined in this article, you can find the vertex of any parabola represented by an equation in standard form. Remember to group the terms, move the constant term to the right side, add and subtract the square of half the coefficient of $x$, factor the perfect square trinomial, divide both sides by the coefficient of the squared term, and add the constant term to both sides. With practice, you will become proficient in finding the vertex of any parabola using the method of completing the square.

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Q: What is the vertex of a parabola?

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A: The vertex of a parabola is the point at which the parabola changes direction. It is a crucial concept in algebra and calculus.

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

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A: To find the vertex of a parabola, you can use the method of completing the square. This involves manipulating the equation to create a perfect square trinomial, which can be factored into the form of $(x - h)^2$.

Q: What is completing the square?

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A: Completing the square is a mathematical technique used to rewrite a quadratic equation in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I complete the square?

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A: To complete the square, follow these steps:

1. Write the equation in standard form.
2. Group the terms.
3. Move the constant term to the right side.
4. Add and subtract the square of half the coefficient of $x$.
5. Factor the perfect square trinomial.
6. Divide both sides by the coefficient of the squared term.
7. Add the constant term to both sides.

Q: What is the formula for the vertex of a parabola?

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A: The formula for the vertex of a parabola is:

$$h = -\frac{b}{2a}$$

$$k = c - \frac{b^2}{4a}$$

Q: How do I use the formula to find the vertex?

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A: To use the formula to find the vertex, plug in the values of $a$, $b$, and $c$ into the formulas for $h$ and $k$.

Q: What if the equation is not in standard form?

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A: If the equation is not in standard form, you will need to rearrange it to write it in standard form before completing the square.

Q: Can I use other methods to find the vertex?

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A: Yes, there are other methods to find the vertex of a parabola, such as using the axis of symmetry or the vertex formula. However, completing the square is a powerful technique that can be used to find the vertex of any parabola.

Q: How do I check my answer?

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A: To check your answer, plug the values of $h$ and $k$ back into the original equation to see if it is true.

Q: What if I get a different answer?

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A: If you get a different answer, double-check your work to make sure you completed the square correctly. If you are still unsure, try using a different method to find the vertex.

Q: Can I use completing the square to find the vertex of a quadratic function?

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A: Yes, completing the square can be used to find the vertex of a quadratic function. However, you will need to rewrite the function in the form of $y = a(x - h)^2 + k$ before finding the vertex.

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

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

• Not writing the equation in standard form
• Not grouping the terms correctly
• Not adding and subtracting the square of half the coefficient of $x$ correctly
• Not factoring the perfect square trinomial correctly
• Not dividing both sides by the coefficient of the squared term correctly
• Not adding the constant term to both sides correctly

Q: How can I practice completing the square?

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A: You can practice completing the square by working through examples and exercises in a textbook or online resource. You can also try completing the square on your own by choosing a quadratic equation and following the steps outlined in this article.

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

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A: Completing the square has many real-world applications, including:

• Finding the vertex of a parabola in physics and engineering
• Modeling population growth and decay in biology
• Analyzing data in statistics
• Solving optimization problems in economics

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

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A: Yes, completing the square can be used to solve systems of equations. However, you will need to rewrite the system of equations in the form of $y = a(x - h)^2 + k$ before solving it.

Q: What are some tips for mastering completing the square?

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A: Some tips for mastering completing the square include:

• Practicing regularly to build your skills and confidence
• Using online resources and video tutorials to supplement your learning
• Working through examples and exercises in a textbook or online resource
• Asking for help from a teacher or tutor if you are struggling
• Reviewing and practicing regularly to maintain your skills.