To answer the question, we need to determine the appropriate name for the formula provided in the problem:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's analyze the options given:
1. Quadratic Formula:
The quadratic formula is specifically designed to find the solutions (roots) of a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. The formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
is indeed used to solve for [tex]\( x \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are coefficients of the quadratic equation.
2. Zero Product Property:
The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers must be zero. Symbolically, if [tex]\( ab = 0 \)[/tex], then either [tex]\( a = 0 \)[/tex] or [tex]\( b = 0 \)[/tex]. This property is used in different contexts, such as factoring quadratic equations, but it isn't the formula given.
3. Quadratic Inequality:
A quadratic inequality is an inequality which involves a quadratic expression, for example, [tex]\( ax^2 + bx + c > 0 \)[/tex]. Solving quadratic inequalities involves different techniques such as finding critical points and testing intervals, but it does not directly relate to the formula in question.
Therefore, the correct choice is:
1. Quadratic Formula
Hence, the provided formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
is known as the Quadratic Formula, and the best answer to the question is option 1, the Quadratic Formula.
