Which of the following quadratic equations has solutions \(x=6 a\) and \(x=-3 b ?\) F. \(x^{2}-18 a b=0\) G. \(x^{2}-x(3 b-6 a)-18 a b=0\) H. \(x^{2}-x(3 b+6 a)+18 a b=0\) J. \(x^{2}+x(3 b+6 a)-18 a b=0\) K. \(x^{2}+x(3 b+6 a)+18 a b=0\)

Vieta's formulas are a set of equations that establish a relationship between the roots of a polynomial and its coefficients. Named after the French mathematician François Viète, these formulas are particularly helpful when dealing with quadratic equations, which take the form \(ax^2 + bx + c = 0\).

In the context of quadratic equations, Vieta's formulas tell us two things: the sum of the roots (\r_1 + r_2\)) and the product of the roots (\r_1 * r_2\). Specifically, for the quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is equal to \( -\frac{b}{a} \) and the product of the roots is \( \frac{c}{a} \).

Understanding Vieta's formulas is crucial because they allow us to quickly write down the relationship between a quadratic equation's coefficients and its roots without solving the equation explicitly. As we shall see, these relationships are key to identifying the equation based on given roots.

The sum of the roots of a quadratic equation is a simple yet powerful concept derived from Vieta's formulas. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the sum is represented as \(r_1 + r_2 = -\frac{b}{a}\).

This relation comes in handy when you know the roots and need to find the coefficient \(b\) of the linear term in the quadratic equation. For instance, if you have two numbers, let's say 2 and 3, and you know these are the roots of a quadratic equation with \(a = 1\), you can instantly determine the equation's linear coefficient will be \(b = -(2 + 3) = -5\) without needing to factor or solve the equation from scratch.

Similarly, the product of the roots of a quadratic equation provides a quick way to determine the constant term \(c\) when the equation is in the standard form \(ax^2 + bx + c = 0\). According to Vieta's formulas, the product of the roots \(r_1 * r_2\) equals \(\frac{c}{a}\).

This is particularly helpful in finding the correct quadratic equation given the roots. For example, if the roots of the equation are 4 and -2, and \(a = 1\), then the product of the roots gives us the constant term \(c\), which is \(4 * (-2) = -8\). This means the quadratic equation with these roots will have a constant term of \(c = -8\).

In the exercise presented, by knowing the roots as 6a and -3b, we immediately deduce the product \(r_1 * r_2\) to be \(6a * (-3b) = -18ab\). This product is the key to matching one of the given options with the correct quadratic equation that incorporates these roots.