Problem 172 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The quadratic formula is developed by applying factoring and the zero-product principle to the quadratic equation \(a x^{2}+b x+c=0\)

Step-by-Step Solution

Answer

The statement is false. The quadratic formula is developed by completing the square method not factoring or using the zero-product principle.

Step 1: Understand the quadratic formula

First, we need to narrate what is the quadratic formula. The quadratic formula is used to find the roots of a quadratic equation and it's given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for any quadratic equation \(ax^2 + bx + c = 0\).

Step 2: Understand the zero-product property

The zero-product property states that if the product of multiple factors is zero, then at least one of the factors must be zero. This property is typically used when factoring quadratic equations to its roots.

Step 3: Evaluate the statement

Now, understanding the quadratic formula and zero-product property, check the given statement. The quadratic formula is actually derived by completing the square, not by factoring or using the zero-product principle. Even though factorization and the zero-product principle can be used to solve some quadratic equations, they are not used to derive the general quadratic formula.

Step 4: Make necessary changes to produce a true statement

To make the statement true, we would need to say: 'The quadratic formula is developed by applying completing the square method to the quadratic equation \(ax^2 + bx + c = 0\)'.