91Ó°ÊÓ

A quadratic equation is any equation that can be rearranged in standard form as:

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are coefficients. The solutions to a quadratic equation, known as roots, depend on these coefficients. A quadratic trinomial may also take a general form as seen in this exercise:

• \(ax^2 + bx + c\)

In our example, the quadratic trinomial is given by \(2y^2 + by + 3\). Finding values that make a trinomial factorable involves determining specific values for these coefficients that lead to integer roots. Understanding the structure of quadratic equations is fundamental to solving various algebraic problems including finding the precise factorable conditions.

The discriminant is a special numeric value that tells you about the nature of the roots of a quadratic equation. For the equation \(ax^2 + bx + c = 0\), the discriminant \(D\) is defined as:

• \(b^2 - 4ac\)

The value of the discriminant provides vital information:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root (a repeated root).
• If \(D < 0\), there are no real roots, only complex numbers.

In the context of our trinomial \(2y^2 + by + 3\), factoring means we need \(b^2 - 24\) to be a perfect square. This ensures that all roots are rational numbers, which makes factoring the trinomial using integer coefficients possible.

A perfect square is an integer that is the square of another integer. For example, 1, 4, 9, 16, and 25 are perfect squares because:

• \(1 = 1^2\)
• \(4 = 2^2\)
• \(9 = 3^2\)
• \(16 = 4^2\)
• \(25 = 5^2\)

In our exercise, we need \(b^2 - 24\) to be a perfect square which ensures that the trinomial is factorable with integer coefficients. By finding such a perfect square value, we can determine a valid \(b\).
When testing integers for \(b^2 = k^2 + 24\), if we find that a specific \(k^2 + 24\) equals a perfect square, it means our required condition is met. In this case, testing \(k=1\) gave \(k^2 + 24 = 25\), which is a perfect square (since \(5^2 = 25\)), leading us to \(b = 5\). These steps help us identify the correct structure needed for factorable trinomials.