Question

In Exercises \(69-72,\) you are tutoring a friend and want to create some quadratic equations that can be solved by factoring. Find a quadratic equation that has the given solutions and explain the procedure you used to obtain the equation. $$-\frac{1}{2}\( and \)\frac{1}{3}$$

Short answer

The formulated quadratic equation is \( 6x^2 + x - 1 = 0 \)

Step-by-step solution

Formulate the Quadratic Equation

Using the given roots, -1/2 and 1/3, and substituting into the generic quadratic format gives: \( a(x - (-1/2))(x - 1/3) = 0 \) which simplifies to \( a(x + 1/2)(x - 1/3) = 0 \)

Simplify the Equation

Expand the equation, ignoring the \( a \) factor for the moment. This results in \( x^2 + 1/2x - 1/3x - 1/6 = 0 \) which simplifies to \( x^2 + 1/6x - 1/6 = 0 \)

Reintroduce the 'a' Factor

Reintroduce the \( a \) factor, choosing a simple integer value to streamline the equation. For example, if \( a \) is chosen as 6, the equation becomes \( 6x^2 + x - 1 = 0 \)