Condition for common roots

An algebraic polynomial can be defined as the algebraic expression that involves many terms of the form cxⁿ, where n is non-negative in nature. For example, 3x³ – 2x² + 5x – 6. A polynomial with its degree 2 can be termed a quadratic polynomial. If f (x) is said to be a quadratic polynomial, then f (x) being equated to 0 is a quadratic equation. The solution of a quadratic equation can be found using various methods such as factoring, square root, completing the square, quadratic formula, graph of a quadratic polynomial. The quadratic formula method is most commonly used. The standard representation of a quadratic equation is ax² + bx + c = 0, where a ≠ 0. The values a, b and c denote real numbers. If a becomes 0, then the equation will reduce to bx + c = 0, which is a linear equation, and not a quadratic equation. The value of the discriminant determines the type of solution or roots. Discriminant can be found using the formula: D = b² – 4ac. There exist 3 different cases:

1] When D > 0: real and distinct roots.

2] When D = 0: real and equal roots.

3] When D < 0: imaginary and roots in pairs.

Illustration: Solve the quadratic equation 6x² + 10x – 1 = 0.

From the above problem, a = 6, b = 10, c = – 1.

To find the value of determinant: D = b² – 4ac is used

D = (10)² – 4 * 6 * (- 1)

= 100 + 24

= 124

D = 124

D > 0

Therefore, from the value of the determinant, we can conclude that the roots are real and distinct.

Using the quadratic formula: x = [- b ± √b² – 4ac] / 2a

= [- 10 ± √10² – 4 * 6 * (- 1)] / (2 * 6)

= [- 10 ± √124] / 12

The two roots are x = [- 10 – √124] / 12 and [- 10 + √124] / 12.

The graph of the quadratic equation is as follows:

The roots of the quadratic equation are given by x = (- b + √discriminant) / 2a and x = (- b – √discriminant) / 2a.

Let these roots be ɑ and β.

The sum of the roots ɑ and β is given by

α + β = (- b + √discriminant) / 2a + (-b – √discriminant) / 2a

= (- b / 2a ) + (√discriminant / 2a) – (b / 2a) – (√discriminant / 2a)

= – 2b / 2a

= – b / a

The product of the roots is as follows:

α * β = (- b + √discriminant) / 2a * (- b – √discriminant) / 2a

= { (- b)² – (√discriminant)² } / (2a)²

= (b² – discriminant) / 4a²

= (b² – b² + 4ac ) /4a² [discriminant = b² – 4ac ]

= c / a

The equation can be formed using the roots. The formula for finding it x² – (sum of roots) x + (product of roots) = 0.

Condition for one and two common root (s)

Consider two quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0.

Let α be the root that is common in the quadratic equations.

Then, a₁α² + b₁α + c₁ = 0 and a₂α² + b₂α + c₂ = 0

Using Cramer’s rule,

The condition for only one common root is (c₁a₂ – c₂a₁)² = (b₁c₂ – b₂c₁) (a₁b₂ – a₂b₁)

If both roots are common, then the condition is (a₁ / a₂) = (b₁ / b₂) = (c₁ / c₂).

Condition for Common Roots – Video Lesson

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