**Condition for Common Root of Quadratic Equations**

A polynomial of the second degree is generally called a quadratic polynomial. In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

If f(x) is a quadratic polynomial, then f(x) = 0 is called a quadratic equation.

An equation in one unknown quantity in the form ax^{2} + bx + c = 0 is called quadratic equation.

**Condition for one common root:**

Let the two quadratic equations are a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0

Now we are going to find the condition that the above quadratic equations may have a common root.

Let α be the common root of the equations a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0. Then,

a_{1}α^{2} + b_{1}α + c_{1} = 0

a^{2}α^{2} + b_{2}α + c_{2} = 0

Now, solving the equations a_{1}α^{2} + b_{1}α + c_{1} = 0, a^{2}α^{2} + b_{2}α + c_{2} = 0 by cross-multiplication, we get

α^{2}/b_{1}c_{2} – b_{2}c_{1} = α/c_{1}a_{2} – c_{2}a_{1} = 1/a_{1}b_{2} – a_{2}b_{1}

⇒ α = b_{1}c_{2} – b_{2}c_{1}/c_{1}a_{2} – c_{2}a_{1}, (From first two)

Or, α = c_{1}a_{2} – c_{2}a_{1}/a_{1}b_{2} – a_{2}b_{1}, (From 2nd and 3rd)

⇒ b_{1}c_{2} – b_{2}c_{1}/c_{1}a_{2} – c_{2}a_{1} = c_{1}a_{2} – c_{2}a_{1}/a_{1}b_{2} – a_{2}b_{1}

⇒ (c_{1}a_{2 }– c_{2}a_{1})^{2} = (b_{1}c_{2} – b_{2}c_{1})(a_{1}b_{2} – a_{2}b_{1}), which is the required condition for one root to be common of two quadratic equations.

The common root is given by α = c_{1}a_{2} – c_{2}a_{1}/a_{1}b_{2} – a_{2}b_{1} or, α = b_{1}c_{2} – b_{2}c_{1}/c_{1}a_{2 }– c_{2}a_{1}

**Note:**

(i) We can find the common root by making the same coefficient of x^{2} of the given equations and then be subtracting the two equations.

(ii) We can find the other root or roots by using the relations between roots and coefficients of the given equations

**Condition for both roots common:**

Let α, β be the common roots of the quadratic equations a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0. Then

α + β = -b1/a1, αβ = c_{1}/a_{1} and α + β = -b_{2}/a_{2}, αβ = c_{2}/a_{2}

Therefore, -b/a_{1} = – b_{2}/a_{2} and c_{1}/a_{1} = c_{2}/a_{2}

⇒ a_{1}/a_{2} = b_{1}/b_{2} and a_{1}/a_{2} = c_{1}/c_{2}

⇒ a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

This is the required condition.

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