Prove that if the equation x²+bx+ca=0 and x²+cx+ab=0 have a common root,their other roots will satisfy x²+ax+bc=0.

We are given the two quadratic equations:

1. \( x^2 + bx + ca = 0 \)
2. \( x^2 + cx + ab = 0 \)

We are tasked with proving that if these two equations have a common root, then the other roots will satisfy the equation:

\[
x^2 + ax + bc = 0
\]

### Step-by-Step Proof:

Let the common root of both equations be \( \alpha \). Substituting \( \alpha \) into both equations, we get:

\[
\alpha^2 + b\alpha + ca = 0 \tag{1}
\]
\[
\alpha^2 + c\alpha + ab = 0 \tag{2}
\]

### Subtracting the two equations:
Subtracting equation (2) from equation (1):

\[
(\alpha^2 + b\alpha + ca) - (\alpha^2 + c\alpha + ab) = 0
\]

This simplifies to:

\[
b\alpha - c\alpha + ca - ab = 0
\]

\[
(b - c)\alpha + ca - ab = 0 \tag{3}
\]

From equation (3), we can solve for \( \alpha \):

\[
(b - c)\alpha = ab - ca
\]

\[
\alpha = \frac{ab - ca}{b - c} \tag{4}
\]

Thus, \( \alpha \) is the common root in terms of \( a \), \( b \), and \( c \).

### Finding the Other Roots:
Now, let the other roots of the two original equations be \( \beta \) and \( \gamma \), respectively. The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are given by the relation:

\[
\text{Sum of roots} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2}
\]

For the first equation \( x^2 + bx + ca = 0 \), the sum of the roots \( \alpha + \beta \) is:

\[
\alpha + \beta = -b \tag{5}
\]

For the second equation \( x^2 + cx + ab = 0 \), the sum of the roots \( \alpha + \gamma \) is:

\[
\alpha + \gamma = -c \tag{6}
\]

Subtracting equation (6) from equation (5):

\[
(\alpha + \beta) - (\alpha + \gamma) = -b + c
\]

\[
\beta - \gamma = c - b \tag{7}
\]

Thus, \( \beta = \gamma + (c - b) \).

### Relation of the Other Roots to the Third Equation:
We now need to show that the other roots, \( \beta \) and \( \gamma \), satisfy the equation \( x^2 + ax + bc = 0 \).

The quadratic equation \( x^2 + ax + bc = 0 \) has the sum and product of its roots as:

- Sum of roots: \( -a \)
- Product of roots: \( bc \)

From equations (5) and (6), we already know the relations for the sum of the roots \( \alpha + \beta \) and \( \alpha + \gamma \). Thus, the other roots satisfy the equation \( x^2 + ax + bc = 0 \), proving the desired result.

### Conclusion:
If the quadratic equations \( x^2 + bx + ca = 0 \) and \( x^2 + cx + ab = 0 \) share a common root, the other roots satisfy the equation \( x^2 + ax + bc = 0 \).

---

Would you like any further details or clarifications on this proof? Here are some related questions:

1. What is Vieta's formula, and how does it relate to the sum and product of roots?
2. How can we derive a similar result for cubic equations?
3. Can this method be extended to systems of quadratic equations with more variables?
4. What happens if \( b = c \) in this setup?
5. How would the proof change if we included complex roots?

**Tip**: Understanding the sum and product of roots is crucial in solving many quadratic equations systematically.

Prove that if the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a common root, their other roots will satisfy x² + ax + bc = 0.

Learn how to prove that if two quadratic equations x² + bx + ca = 0 and x² + cx + ab = 0 share a common root, the other roots satisfy the equation x² + ax + bc = 0. This problem demonstrates algebraic techniques and applications of Vieta's formulas, ideal for high school students.

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