Understanding "Both Roots Common" in Quadratic Equations

When we talk about "both roots common" in the context of quadratic equations, we are referring to a scenario where two quadratic equations share the same set of solutions. This concept is important in algebra and is often encountered in problems involving systems of equations.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form:

$$ ax^2 + bx + c = 0 $$

where (a), (b), and (c) are constants, and (a \neq 0). The solutions to this equation, also known as the roots, can be found using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Conditions for Both Roots Common

For two quadratic equations to have both roots in common, they must satisfy certain conditions. Let's consider two quadratic equations:

1. ( ax^2 + bx + c = 0 ) (Equation 1)
2. ( dx^2 + ex + f = 0 ) (Equation 2)

For both roots to be common, the following must hold:

• The ratios of the corresponding coefficients of the two equations must be equal, i.e.,

$$ \frac{a}{d} = \frac{b}{e} = \frac{c}{f} $$

This implies that the two equations are essentially the same, just scaled versions of one another.

Table of Differences and Important Points

Aspect	Single Root Common	Both Roots Common
Coefficient Relationship	Only two ratios are equal	All three ratios must be equal
Nature of Equations	Equations are different	Equations are scaled versions
Number of Common Roots	One	Two
Example	(x^2 - 5x + 6 = 0) and (2x^2 - 10x + 12 = 0) (3 is a common root)	(x^2 - 5x + 6 = 0) and (2x^2 - 10x + 12 = 0) (Both 2 and 3 are common roots)

Examples

Example 1: Identifying Both Roots Common

Consider the following equations:

1. ( x^2 - 5x + 6 = 0 ) (Equation 1)
2. ( 2x^2 - 10x + 12 = 0 ) (Equation 2)

Let's check the ratios of the corresponding coefficients:

$$ \frac{1}{2} = \frac{-5}{-10} = \frac{6}{12} $$

Since all the ratios are equal, both roots of Equation 1 are also roots of Equation 2.

Example 2: Finding Common Roots

Given two equations:

1. ( x^2 - 3x + 2 = 0 ) (Equation 1)
2. ( 2x^2 - 6x + 4 = 0 ) (Equation 2)

We can see that:

$$ \frac{1}{2} = \frac{-3}{-6} = \frac{2}{4} $$

Thus, both roots of Equation 1 are common with Equation 2. We can find the common roots by solving either equation:

$$ x^2 - 3x + 2 = 0 $$

Using the quadratic formula or factoring, we find the roots to be ( x = 1 ) and ( x = 2 ).

Conclusion

Understanding the concept of "both roots common" is crucial when dealing with systems of quadratic equations. It allows us to determine whether two equations are essentially the same and to find their common solutions. Remember that for both roots to be common, the ratios of the corresponding coefficients must be equal across the two equations.