Final answer:
To determine if an equation with x and y is quadratic, ensure it can be written in or resembles the form ax²+bx+c = 0, where a is nonzero. This form is characteristic of quadratic, or second-order, polynomials. The quadratic formula is used to find the solutions to such equations.
Step-by-step explanation:
To determine if an equation in variables x and y is quadratic, look for the presence of a second-order term, which is an x with an exponent of 2 (x²). A general form of a quadratic equation is ax²+bx+c = 0, where a, b, and c are constants, and a is not equal to zero. If an equation follows this format or can be rearranged into this format, it is considered quadratic. The quadratic formula, x = [-b ± √(b²-4ac)]/(2a), is used to solve for the roots of the quadratic equation.
For example, the equation y = ax² + bx + c represents a quadratic function, where y is the dependent variable and x is the independent variable. This function is known as a second-order polynomial or a quadratic function. When faced with an equation where the highest power of x is squared, such as in y = 3x² + 2x - 5, the equation is clearly quadratic.