#1**0 **

There seem to be a lot of questions about families of quadratics and their respective number of roots!

I will write the given quadratic equation in standard form.

\(3x^2 + 7x + c = 15x - 10 \\ 3x^2 - 8x + (c + 10) = 0\)

In order for this family of quadratics to have 2 real roots, the discriminant of the quadratic must be greater than zero.

\(a = 3; b = -8; c = c + 10 \\ D = b^2 - 4ac \\ D = (-8)^2 - 4 * 3 * (c + 10) \\ D = 64 - 12(c + 10) \\ D = -12c -56\)

Now that I have simplified the discriminant as much as possible, I will determine what values of c make the discriminant greater than 0.

\(D > 0 \\ -12c - 56 > 0 \\ -12c > 56 \\ c < -\frac{14}{3}\)

There are no positive integers c such that the given quadratic has two real roots! Therefore, I do not think it makes sense to take the product!

The3Mathketeers Aug 30, 2023