If the quadratic equations x ^ 2 + px + q = 0 and x ^ 2 + p' * x + q' = 0 have a common root show it must be either (p * q' - p'q)/(q - q') or (q - q')/(p'- p)


Given that the quadratic equations:

1. \( x^2 + px + q = 0 \)
2. \( x^2 + p'x + q' = 0 \)

have a common root, let's denote this common root by \( \alpha \). 

### Step-by-step Solution:

#### 1. Substitution of the Common Root:

Since \( \alpha \) is a root of both equations, we have:

\[
\alpha^2 + p\alpha + q = 0 \quad \text{(Equation 1)}
\]
\[
\alpha^2 + p'\alpha + q' = 0 \quad \text{(Equation 2)}
\]

#### 2. Subtract the Equations:

Subtract Equation 2 from Equation 1 to eliminate \( \alpha^2 \):

\[
(\alpha^2 + p\alpha + q) - (\alpha^2 + p'\alpha + q') = 0
\]

Simplifying this gives:

\[
(p - p') \alpha + (q - q') = 0
\]

Rearrange to solve for \( \alpha \):

\[
(p - p') \alpha = -(q - q')
\]

\[
\alpha = \frac{q - q'}{p' - p}
\]

#### 3. Express \( \alpha \) in Another Form:

Alternatively, subtract Equation 1 from Equation 2:

\[
(\alpha^2 + p'\alpha + q') - (\alpha^2 + p\alpha + q) = 0
\]

Simplifying this gives:

\[
(p' - p) \alpha + (q' - q) = 0
\]

Rearrange to solve for \( \alpha \):

\[
(p' - p) \alpha = -(q' - q)
\]

\[
\alpha = \frac{q' - q}{p - p'}
\]

Notice that this is equivalent to:

\[
\alpha = \frac{q - q'}{p' - p}
\]

Thus, we can say:

\[
\alpha = \frac{p q' - p'q}{q - q'} \quad \text{or} \quad \alpha = \frac{q - q'}{p' - p}
\]

This completes the proof that the common root must be either:

1. \( \alpha = \frac{p q' - p'q}{q - q'} \)
2. \( \alpha = \frac{q - q'}{p' - p} \)

### Conclusion:

Both expressions for the common root are valid depending on the form in which we express the terms.

Would you like a more detailed explanation of any particular step or have any further questions?

### Follow-Up Questions:

1. What happens if \( q = q' \)? Can we still find a common root?
2. How does the discriminant of each quadratic equation relate to the common root?
3. Can we generalize this result to higher-degree polynomials?
4. How can we verify the common root if we know the values of \( p, p', q, \) and \( q' \)?
5. What role does Vieta's formulas play in finding the common root?

#### Tip:

When two polynomials share a common root, expressing the polynomials in terms of their coefficients and manipulating the equations can often reveal relationships between the coefficients and the common root.


Understand how quadratic equations share common roots through detailed examples and proofs. Explore the relationship between coefficients and common roots in quadratic equations. Suitable for high school students and math enthusiasts.

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