Which equation has two real, distinct solutions? Do not actually solve. A. (3x-4)² = -9 B. (4-7x)² = 0 C. (5x-9)(5x-9) = 0 D. (7x+4)² = 11

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or the quadratic formula. The nature of the solutions—real and distinct, real and repeated, or complex—depends on the discriminant (b² - 4ac).

Discriminant

The discriminant is a key component in determining the nature of the roots of a quadratic equation. It is calculated as b² - 4ac. If the discriminant is positive, the equation has two distinct real solutions; if it is zero, there is one repeated real solution; and if it is negative, the solutions are complex. Understanding the discriminant helps in quickly assessing the type of solutions without solving the equation.

Perfect Squares

A perfect square is an expression that can be written as the square of a binomial, such as (a ± b)² = a² ± 2ab + b². In the context of quadratic equations, recognizing perfect squares can simplify the analysis of solutions. For example, if an equation is set equal to zero and is a perfect square, it indicates that the solutions may be repeated, affecting the count of distinct solutions.