To determine which of the given options correctly specifies the highest degree in an equation where the quadratic formula can be used to solve it, we need to understand the nature of quadratic equations and the quadratic formula itself.
A quadratic equation is typically written in the standard form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where:
- [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants,
- [tex]\( a \neq 0 \)[/tex],
- and [tex]\( x \)[/tex] represents the variable.
The quadratic formula that solves the equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
The key characteristic to recognize here is that the term [tex]\( ax^2 \)[/tex] dictates that the highest exponent (or degree) of the variable [tex]\( x \)[/tex] in this equation is 2. Therefore, the quadratic formula specifically addresses equations where the highest degree is 2.
Let us now examine each of the provided options:
- Option A: [tex]\( 0 \)[/tex] – This relates to equations that are constants, taking the form [tex]\( c = 0 \)[/tex]. Such an equation cannot be solved using the quadratic formula.
- Option B: [tex]\( 1 \)[/tex] – This corresponds to linear equations of the form [tex]\( ax + b = 0 \)[/tex]. These equations are solved using simple algebra and do not require the quadratic formula.
- Option C: [tex]\( 2 \)[/tex] – This is indeed the highest degree for which the quadratic formula is designed. Quadratic equations have this degree.
- Option D: [tex]\( 3 \)[/tex] – This represents cubic equations, which take the form [tex]\( ax^3 + bx^2 + cx + d = 0 \)[/tex]. They are solved using methods different from the quadratic formula.
Hence, the quadratic formula can be used to solve an equation only if the highest degree in the equation is:
○ C. 2
