# Common Roots

## Common Roots

Condition for One Common Roots: Let us find the condition such that quadratic equation a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 may have a common root. Let α be the common root of given equations.

Then, a₁α² + b₁α + c₁ = 0 and a₂α ² + b₂α + c₂ = 0.

Solving these two equations by cross – multiplication, we have

$$\frac{{{\alpha }^{2}}}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\frac{\alpha }{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}=\frac{1}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$$,

$${{\alpha }^{2}}=\frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$$,

$$\alpha =\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$$,

$$\frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}={{\left( \frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)}^{2}}$$,

(c₁a₂ – c₂a₁)² = (b₁c₂ – b₂c₁) (a₁b₂ – a₂b₁)

This condition can easily be remembered by cross multiplication method

This is the condition required for a root to be common to two quadratic equation. The common roots are given by $$\alpha =\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$$.

Example: Determine the values of m for which equation 33x² + 4mx + 2 = 0 and 2x² + 3x – 2 = 0 may have a common root.

Solution: Let α be the common root of the equation 3x² + 4mx + 2 = 0 and 2x² + 3x – 2 = 0.

Then α must satisfy both the equation. Therefore

3α² + 4mα + 2 = 0

2α² + 3α – 2 = 0

$$\frac{{{\alpha }^{2}}}{-8m-6}=\frac{-\alpha }{-6-4}=\frac{1}{9-8m}$$,

(-6 – 4)² = (9 – 8m) (-8m – 6)

m = -11/8, 7/4.