Certainly! Let's break down the expression [tex]\(9a^2 + 10a + 1\)[/tex] step by step.
### Understanding the Expression
We have a quadratic expression:
[tex]\[ 9a^2 + 10a + 1 \][/tex]
This is already in its simplified form. To understand it better, let's identify the components:
1. Coefficients:
- The coefficient of [tex]\(a^2\)[/tex] is 9.
- The coefficient of [tex]\(a\)[/tex] is 10.
- The constant term is 1.
2. Standard Form:
- This expression is in the standard quadratic form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\(a = 9\)[/tex]
- [tex]\(b = 10\)[/tex]
- [tex]\(c = 1\)[/tex]
### Detailed Examination
#### 1. Quadratic Term ([tex]\(9a^2\)[/tex]):
- This term represents the portion of the expression that changes with the square of [tex]\(a\)[/tex]. The higher the value of [tex]\(a\)[/tex], the more rapidly [tex]\(9a^2\)[/tex] will increase since it grows proportionally to [tex]\(a^2\)[/tex].
#### 2. Linear Term ([tex]\(10a\)[/tex]):
- This term changes linearly with [tex]\(a\)[/tex]. Each unit increase in [tex]\(a\)[/tex] adds 10 to the value of the expression.
#### 3. Constant Term (1):
- This term does not change with [tex]\(a\)[/tex]; it simply adds 1 to the overall value of the expression.
### Importance of the Expression
- Graph Representation: The graph of this quadratic expression will be a parabola facing upwards since the coefficient of [tex]\(a^2\)[/tex] (which is 9) is positive.
- Roots/Solutions: To find the roots of the quadratic expression, one typically sets the expression equal to zero and solves for [tex]\(a\)[/tex]. For this specific expression, you would solve the equation [tex]\(9a^2 + 10a + 1 = 0\)[/tex] using methods such as factoring, completing the square, or using the quadratic formula.
### Conclusion
The given quadratic expression is in its simplest form and comprises three terms: a quadratic term [tex]\(9a^2\)[/tex], a linear term [tex]\(10a\)[/tex], and a constant term 1. The expression does not need any further simplification.
So,
[tex]\[ 9a^2 + 10a + 1 \][/tex]
is the detailed form of the quadratic expression.
