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:

ax2+ bx + c = 0

px2 + 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:

ax2 + bx + c = 0

px2 + qx + r = 0

  • A common root α must satisfy both equations when you substitute it in:

aα2 + bα + c = 0

pα2 + 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:

aα2 + bα + c = pα2 + 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)α2 + (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:

2x2 - 5x + 3 = 0

3x2 - 7x + 2 = 0

  • To find if they have a common root, subtract one equation from the other:

(2 - 3)x2 + (-5 + 7)x + (3 - 2) = 0

-x2+ 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.