Sure, let's analyze the given equation step by step:
The equation given is of the form [tex]\(a x^2 + b x + c = y\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are real numbers, and [tex]\(a \neq 0\)[/tex].
1. Identify the elements:
- The term [tex]\(a x^2\)[/tex] is a squared term and it indicates that the equation involves a quadratic term.
- The term [tex]\(b x\)[/tex] is a linear term.
- The term [tex]\(c\)[/tex] is a constant.
- The equation equals [tex]\(y\)[/tex], which means it has a standard quadratic form.
2. Determine the characteristics:
- An equation of the form [tex]\(a x^2 + b x + c = 0\)[/tex] is known as a quadratic equation.
- A quadratic equation specifically refers to equations involving powers of [tex]\(x\)[/tex] up to 2 (i.e., [tex]\(x^2\)[/tex] is the highest power).
- The coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are real numbers.
- The condition [tex]\(a \neq 0\)[/tex] is crucial because if [tex]\(a\)[/tex] were 0, the equation would reduce to a linear equation.
3. Comparison with the options:
- Quadratic equation: This directly fits the form [tex]\(a x^2 + b x + c = y\)[/tex] since [tex]\(a \neq 0\)[/tex].
- Quadratic inequality: This would be in the form of [tex]\(a x^2 + b x + c \leq y\)[/tex] or [tex]\(a x^2 + b x + c \geq y\)[/tex], etc.
- Zero Product Property: This is a property used in the context of solving quadratic equations but it is not a type of equation itself.
Based on the analysis, the equation [tex]\(a x^2 + b x + c = y\)[/tex] is best described as a:
Quadratic equation
Therefore, the best choice from the given options is Quadratic equation.
