To analyze the quadratic expression [tex]\( x^2 + 5x + 3 \)[/tex], we can proceed with a detailed, step-by-step breakdown.
### Step 1: Identify the quadratic expression
The given quadratic expression is:
[tex]\[
x^2 + 5x + 3
\][/tex]
### Step 2: Recognize the general form of a quadratic equation
The general form of a quadratic equation is:
[tex]\[
ax^2 + bx + c
\][/tex]
where:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex]
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]
- [tex]\( c \)[/tex] is the constant term
For our specific quadratic expression [tex]\( x^2 + 5x + 3 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 3: Write the expression
The expression can be written directly as:
[tex]\[
x^2 + 5x + 3
\][/tex]
### Step 4: Check the behavior of the expression
- Since [tex]\( a = 1 \)[/tex] which is positive, the parabola opens upwards.
- The coefficients of the expression tell us about various properties such as the direction of the parabola and the vertex.
### Step 5: Determine the roots (optional step for comprehensive understanding)
If you want to find the roots of the quadratic expression, you would use the quadratic formula given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Applying the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{-5 \pm \sqrt{25 - 12}}{2} = \frac{-5 \pm \sqrt{13}}{2}
\][/tex]
Thus, the roots of the quadratic equation [tex]\( x^2 + 5x + 3 \)[/tex] are:
[tex]\[
x = \frac{-5 + \sqrt{13}}{2} \quad \text{and} \quad x = \frac{-5 - \sqrt{13}}{2}
\][/tex]
### Conclusion
To summarize, the given expression is [tex]\( x^2 + 5x + 3 \)[/tex], which represents a quadratic equation with:
- Leading coefficient: [tex]\( a = 1 \)[/tex]
- Linear coefficient: [tex]\( b = 5 \)[/tex]
- Constant term: [tex]\( c = 3 \)[/tex]
The roots, derived from the quadratic formula, are optional extra information but help deepen understanding of the expression's behavior.
