1. Let's state the problem: We want to understand why in the quadratic formula, the product of the denominators of the roots equals $a$ when solving $ax^2 + bx + c = 0$. 2. The quadratic formula for the roots is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. Notice that the denominator for both roots is $2a$. This means both roots share the same denominator. 4. The question asks why the product of the denominators equals $a$. Since both denominators are $2a$, their product is: $$ (2a) \times (2a) = 4a^2 $$ 5. However, the question might be referring to the product of the roots themselves, not the denominators. The product of the roots $x_1$ and $x_2$ is: $$ x_1 \times x_2 = \frac{c}{a} $$ 6. This comes from the factorization of the quadratic equation and the relationship between coefficients and roots: - Sum of roots: $x_1 + x_2 = -\frac{b}{a}$ - Product of roots: $x_1 x_2 = \frac{c}{a}$ 7. So, the denominator $a$ appears in the product of the roots, not the denominators of the roots themselves. 8. To summarize, the denominators of the roots are both $2a$, and their product is $4a^2$. The product of the roots themselves is $\frac{c}{a}$, which involves $a$ in the denominator. This explains the role of $a$ in the quadratic formula denominators and the roots' product.