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:

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:

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.