Identify the vertex, axis of symmetry, and x-intercept(s). vertex (x, g(x)) = axis of symmetry x-intercept (x, g(x)) =
Identify the vertex, axis of symmetry, and x-intercept(s). vertex (x, g(x)) = axis of symmetry x-intercept (x, g(x)) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Identify the vertex, axis of symmetry, and x-intercept(s).
| vertex |
(x, g(x)) =
|
|
||
| axis of symmetry |
|
|||
| x-intercept |
(x, g(x)) =
|
|
![### Writing the Quadratic Function in Standard Form
To write the quadratic function in standard form, we start with the given equation and complete the following steps:
Given:
\[ g(x) = x^2 + 10x + 25 \]
We can rewrite this quadratic function by recognizing a perfect square trinomial:
\[ g(x) = (x + 5)^2 \]
Therefore, the standard form of the quadratic function is:
\[ g(x) = (x + 5)^2 \]
Note: The green checkmark indicates that this answer is correct.
### Explanation of Completing the Square
In this case, the given quadratic expression \( x^2 + 10x + 25 \) can be observed as a perfect square because it fits the form \( (x + a)^2 \), where 'a' is a constant.
When expanded, \( (x + a)^2 = x^2 + 2ax + a^2 \). Here, a is 5 because \( 2 \cdot 5 = 10 \) and \( 5^2 = 25 \).
We have:
- The term \( x^2 \)
- The term \( 10x \) which is \( 2 \cdot 5 \cdot x \)
- The term \( 25 \) which is \( 5^2 \)
Thus, we can factor the quadratic expression as \( (x + 5)^2 \).
This illustrates the method of completing the square and recognizing perfect square trinomials.
---
Feel free to include additional insights or related examples to further illustrate the concept if needed for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2F72107c9f-8f2c-49c1-857a-a16c385ab735%2Fdybqprv_processed.png&w=3840&q=75)
Transcribed Image Text:### Writing the Quadratic Function in Standard Form
To write the quadratic function in standard form, we start with the given equation and complete the following steps:
Given:
\[ g(x) = x^2 + 10x + 25 \]
We can rewrite this quadratic function by recognizing a perfect square trinomial:
\[ g(x) = (x + 5)^2 \]
Therefore, the standard form of the quadratic function is:
\[ g(x) = (x + 5)^2 \]
Note: The green checkmark indicates that this answer is correct.
### Explanation of Completing the Square
In this case, the given quadratic expression \( x^2 + 10x + 25 \) can be observed as a perfect square because it fits the form \( (x + a)^2 \), where 'a' is a constant.
When expanded, \( (x + a)^2 = x^2 + 2ax + a^2 \). Here, a is 5 because \( 2 \cdot 5 = 10 \) and \( 5^2 = 25 \).
We have:
- The term \( x^2 \)
- The term \( 10x \) which is \( 2 \cdot 5 \cdot x \)
- The term \( 25 \) which is \( 5^2 \)
Thus, we can factor the quadratic expression as \( (x + 5)^2 \).
This illustrates the method of completing the square and recognizing perfect square trinomials.
---
Feel free to include additional insights or related examples to further illustrate the concept if needed for educational purposes.
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