Understanding the Concept of "One Common Root" in Quadratic Equations

When we talk about quadratic equations, we are referring to equations of the form:

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

where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). A quadratic equation typically has two roots, which can be real or complex numbers. These roots are the solutions to the equation, i.e., the values of ( x ) that satisfy the equation.

One Common Root

The concept of "one common root" comes into play when we have two quadratic equations and they share exactly one root. This means that there is one value of ( x ) that satisfies both equations. Let's consider two quadratic equations:

$$ \begin{align*} a_1x^2 + b_1x + c_1 &= 0 \quad \text{(Equation 1)} \ a_2x^2 + b_2x + c_2 &= 0 \quad \text{(Equation 2)} \end{align*} $$

If these two equations have one common root, let's denote it by ( \alpha ). Then ( \alpha ) satisfies both equations:

$$ \begin{align*} a_1\alpha^2 + b_1\alpha + c_1 &= 0 \ a_2\alpha^2 + b_2\alpha + c_2 &= 0 \end{align*} $$

Condition for One Common Root

For two quadratic equations to have one common root, their coefficients must satisfy a certain relationship. This relationship is derived from the fact that the product of the roots of a quadratic equation is equal to ( \frac{c}{a} ) and the sum of the roots is equal to ( -\frac{b}{a} ).

If ( \alpha ) is the common root and ( \beta ) and ( \gamma ) are the other roots of Equation 1 and Equation 2, respectively, then:

$$ \begin{align*} \alpha + \beta &= -\frac{b_1}{a_1} \ \alpha + \gamma &= -\frac{b_2}{a_2} \ \alpha\beta &= \frac{c_1}{a_1} \ \alpha\gamma &= \frac{c_2}{a_2} \end{align*} $$

By equating the sums and products of the roots, we can derive the condition for one common root.

Example

Let's consider two quadratic equations:

$$ \begin{align*} x^2 - 5x + 6 &= 0 \quad \text{(Equation 1)} \ 2x^2 - 7x + k &= 0 \quad \text{(Equation 2)} \end{align*} $$

To find the value of ( k ) for which there is one common root, we can use the condition for one common root. Let's assume that the common root is ( \alpha ). Then:

$$ \begin{align*} \alpha^2 - 5\alpha + 6 &= 0 \ 2\alpha^2 - 7\alpha + k &= 0 \end{align*} $$

By subtracting the first equation from the second (after multiplying the first by 2), we get:

$$ \alpha^2 - 3\alpha + (k - 12) = 0 $$

For ( \alpha ) to be a common root, ( k - 12 ) must be zero, so ( k = 12 ).

Differences and Important Points

Aspect	Without Common Root	With One Common Root
Number of Common Roots	0	1
Coefficient Relationship	No specific relation	Must satisfy a condition derived from the sum and product of roots
Example Equations	( x^2 - 4x + 3 = 0 ) and ( x^2 - x - 6 = 0 )	( x^2 - 5x + 6 = 0 ) and ( 2x^2 - 7x + 12 = 0 )
Example Roots	( {1, 3} ) and ( {-2, 3} )	( {2, 3} ) and ( {3, 4} ) (assuming ( k = 12 ))
Graphical Representation	Two parabolas with no intersection points	Two parabolas intersecting at exactly one point

In summary, when two quadratic equations have one common root, there is a specific relationship between their coefficients. This relationship can be used to find the common root or to determine the conditions under which the common root exists. Understanding this concept is crucial for solving problems involving quadratic equations, especially in algebra and pre-calculus.