To determine which quadratic expression is written in standard form, we need to understand what constitutes a standard form for a quadratic expression. The standard form of a quadratic expression is given by:
[tex]\[ ax^2 + bx + c \][/tex]
Where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable.
Let's analyze each option:
### Option A: [tex]\( x^2 - 4x + 2 \)[/tex]
This expression is already of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 2 \)[/tex]. Therefore, this expression is in standard form.
### Option B: [tex]\( (x + 8)^2 \)[/tex]
To check if this is in standard form, we need to expand it:
[tex]\[ (x + 8)^2 = (x + 8)(x + 8) = x^2 + 16x + 64 \][/tex]
Now the expression [tex]\( x^2 + 16x + 64 \)[/tex] is indeed in standard form, but the original form [tex]\( (x + 8)^2 \)[/tex] is not directly written as [tex]\( ax^2 + bx + c \)[/tex].
### Option C: [tex]\( (x + 4)x \)[/tex]
To check if this is in standard form, we need to expand it:
[tex]\[ (x + 4)x = x^2 + 4x \][/tex]
The expanded form [tex]\( x^2 + 4x \)[/tex] is in standard form where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 0 \)[/tex]. Therefore, once expanded, this expression fits the standard form.
### Option D: [tex]\( x^2 + 7(x + 2) \)[/tex]
To check if this is in standard form, we need to expand it:
[tex]\[ x^2 + 7(x + 2) = x^2 + 7x + 14 \][/tex]
The expanded form [tex]\( x^2 + 7x + 14 \)[/tex] is indeed in standard form where [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = 14 \)[/tex].
### Conclusion
- Option A ([tex]\( x^2 - 4x + 2 \)[/tex]) is in standard form.
- Option B ([tex]\( (x + 8)^2 \)[/tex]) is not directly in standard form until expanded.
- Option C ([tex]\( (x + 4)x \)[/tex]) is in standard form when expanded.
- Option D ([tex]\( x^2 + 7(x + 2) \)[/tex]) is in standard form when expanded.
Thus, Options A, C, and D are quadratic expressions written in standard form.
