To determine which expression represents a quadratic expression, let's first understand what a quadratic expression is. A quadratic expression is a polynomial of degree 2, which means the highest power of [tex]\( x \)[/tex] in the expression is [tex]\( x^2 \)[/tex].
Now, we'll analyze each given expression to see which one meets the criteria of being a quadratic expression.
1. [tex]\( 6x^4 - 5x^3 + 3x^2 - 7x - 8 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^4 \)[/tex].
- This is a polynomial of degree 4, not a quadratic expression.
2. [tex]\( 5x^3 + 3x^2 - 7x - 8 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex].
- This is a polynomial of degree 3, not a quadratic expression.
3. [tex]\( 2x^2 + 3x - 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
- This is a polynomial of degree 2, which is indeed a quadratic expression.
4. [tex]\( 3x - 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex].
- This is a polynomial of degree 1, not a quadratic expression.
Based on this analysis, the expression that represents a quadratic expression is:
[tex]\[ 2x^2 + 3x - 1 \][/tex]
To confirm our answer, we'll identify the index (position) of this expression in the given list:
1. [tex]\( 6x^4 - 5x^3 + 3x^2 - 7x - 8 \)[/tex] - Index 0
2. [tex]\( 5x^3 + 3x^2 - 7x - 8 \)[/tex] - Index 1
3. [tex]\( 2x^2 + 3x - 1 \)[/tex] - Index 2
4. [tex]\( 3x - 1 \)[/tex] - Index 3
The quadratic expression, [tex]\( 2x^2 + 3x - 1 \)[/tex], is at index 2 in the list. Thus, the result is:
[tex]\[ (2, \['2x^2 + 3x - 1'\][/tex]) \]
So, the correct answer is:
[tex]\[ 2x^2 + 3x - 1 \][/tex]
