Condition for two Quadratic Equations to have a Common Root
In algebra, two quadratic equations can have a common root under certain conditions. The common root is a value of x that satisfies both equations. To determine when this occurs, consider the following:
1. Two Quadratic Equations:
 Start with two quadratic equations in the general form:
a${x}^{2}$+ bx + c = 0
p${x}^{2}$ + qx + r = 0
 Where a, b, c, p, q, and r are constants, and a and p are not equal to zero.
2. Form of a Common Root:
 A common root is a value of x that satisfies both equations simultaneously. This means it must be a solution for both equations:
a${x}^{2}$ + bx + c = 0
p${x}^{2}$ + qx + r = 0
 A common root α must satisfy both equations when you substitute it in:
a${\alpha}^{2}$ + bα + c = 0
p${\alpha}^{2}$ + qα + r = 0
3. Equality of the Common Root:
 For two quadratic equations to have a common root, the common root α must be the same in both equations. In other words:
a${\alpha}^{2}$ + bα + c = p${\alpha}^{2}$ + qα + r
 When you have this equality, it means that α satisfies both equations, and thus, the two equations have a common root.
4. Condition for Common Roots:
 To express this condition more clearly, you can subtract one equation from the other:
(a  p)${\alpha}^{2}$ + (b  q)α + (c  r) = 0

If the above equation has a real solution for α, it indicates that the two quadratic equations have a common real root.

If it has a complex solution for α, it means that the two equations have a common complex root.
5. Example:
 Consider two quadratic equations:
2${x}^{2}$  5x + 3 = 0
3${x}^{2}$  7x + 2 = 0
 To find if they have a common root, subtract one equation from the other:
(2  3)${x}^{2}$ + (5 + 7)x + (3  2) = 0
${x}^{2}$+ 2x + 1 = 0
 Solving this equation, you find that x = 1 is a common root for both equations.
6. Summary:

For two quadratic equations to have a common root, the common root must satisfy both equations simultaneously, meaning it is a solution for both.

Expressing this condition mathematically, you set both equations equal to each other and solve for the common root α. If it has a real or complex solution, the two equations have a common root.