Condition of both common roots of a pair of quadratic equations

Comparing Roots of Two Quadratic Equations

Given:
Two quadratic equations:
First equation: \( a_1x^2 + b_1x + c_1 = 0 \)
Second equation: \( a_2x^2 + b_2x + c_2 = 0 \)

Let the roots of equation (1) be \( \alpha \) and \( \beta \).
By Vieta’s formulas:

Sum of roots:   \( \alpha + \beta = -\frac{b_1}{a_1} \)

The product of the roots can be expressed as \( \alpha \beta = \frac{c_1}{a_1} \)

Similarly, for equation (2), if it has the same roots \( \alpha \) and \( \beta \), then:

Sum of roots:   \( \alpha + \beta = -\frac{b_2}{a_2} \)
Product of roots:   \( \alpha \cdot \beta = \frac{c_2}{a_2} \)

Now Compare:

• If both the sums of roots are equal: \( \frac{b_1}{a_1} = \frac{b_2}{a_2} \)
• And both the products of roots are equal: \( \frac{c_1}{a_1} = \frac{c_2}{a_2} \)

Then the equations share the same roots. This leads to the proportionality condition:

\( frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

If the sums and products of roots of both equations match as shown, the roots are the same. Otherwise, they are not.