JEE Main 2024 Question Paper Solution Discussion Live JEE Main 2024 Question Paper Solution Discussion Live

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:

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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:

Condition for Common Roots

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,

\(\begin{array}{l}\frac{\alpha^{2}}{\begin{vmatrix} -c_{1} &b_{1} \\ -c_{2} &b_{2} \end{vmatrix}}=\frac{\alpha}{\begin{vmatrix} a_{1} &-c_{1} \\ a_{2} &-c_{2} \end{vmatrix}}=\frac{1}{\begin{vmatrix} a_{1} &b_{1} \\ a_{2} &b_{2} \end{vmatrix}}\end{array} \)

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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Frequently Asked Questions

Q1

What do you mean by a quadratic equation?

An equation of the form ax2+bx+c=0, where a, b, c are real numbers and a ≠ 0.

Q2

What do you mean by roots of a quadratic equation?

The values of variables satisfying the given quadratic equation are called the roots of the quadratic equation.

Q3

Give the condition for real and distinct roots of a quadratic equation.

If the discriminant, D > 0, then the roots are real and distinct.

Q4

How to calculate the discriminant of a quadratic equation?

Discriminant is given by the formula, D = (b2 – 4ac).

Q5

Give the condition for real and equal roots of a quadratic equation.

If the discriminant, D = 0, then the roots are real and equal.

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