Final answer:
The equation y = -2(x - 3)^2 – 5 is a quadratic equation because it has a squared term. The equation 7x^2 + 3y = 12 is quadratic in x due to the x^2 term. The equation 20x^2 + 3y^2 = 12, while not a standard one-variable quadratic equation, also falls under the family of quadratics since both x and y are squared.
Step-by-step explanation:
The equations to be analyzed for their quadratic nature are y = -2(x - 3)^2 – 5, 7x^2 + 3y = 12, and 20x^2 + 3y^2 = 12. A quadratic equation is an equation of the second degree, which means it includes at least one term that is squared. The standard form of a quadratic equation in one variable is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0.
The first equation, y = -2(x - 3)^2 – 5, is a quadratic equation because it has the term (x - 3)^2, which is the square of a binomial, indicating that x is raised to the second power.
The second equation, 7x^2 + 3y = 12, is a quadratic equation in x because it includes the x^2 term.
The third equation, 20x^2 + 3y^2 = 12, is not a standard quadratic equation in one variable, but it is a quadratic in two variables because both x and y are squared. It is therefore part of the family of quadratic equations, but not a pure quadratic equation in the sense of having a single variable squared.