Consider quadratic polynomial P(x) = 2x^{2} – 16x + 30.

Now, 2x^{2} – 16x + 30

= (2x – 6) (x – 3)

= 2(x – 3) (x – 5)

The zeroes of P(x) are 3 and 5.

**Sum of the zeroes are **= 3 + 5 = 8 ==

**Product of the zeroes are **= 3 × 5 = 15 = =

So if ax^{2} + bx + c, a ≠ 0 is a quadratic polynomial and a, b are two zeroes of polynomial then

Example: Find the zeroes of the quadratic polynomial 6x^{2}– 13x + 6 and verify the relation between the zeroes and its coefficients.

Solution: We have,

6x^{2} – 13x + 6

= 6x^{2} – 4x – 9x + 6

= 2x (3x – 2) –3 (3x – 2)

= (3x – 2) (2x – 3)

So, the value of quadratic equation 6x^{2} – 13x + 6 is 0, when

(3x – 2) = 0 or (2x – 3) = 0 i.e.,

When x = or

Therefore, the zeroes of 6x^{2} – 13x + 6 are and .

**Sum of the zeroes** :

+ = = =

**Product of the zeroes**

= × = =

Example: Find the zeroes of the quadratic polynomial 4x2 – 9 and verify the relation between the zeroes and its coefficients.

Solution: We have,

4x^{2} – 9

= (2x)^{2} – 3^{2}

= (2x – 3) (2x + 3)

So, the value ofa quadratic equation 4x^{2} – 9 is 0, when

2x – 3 = 0 or 2x + 3 = 0

i.e., when x = or x = .

Therefore, the zeroes of 4x^{2} – 9 are & .

**Sum of the zeroes **

= + = 0 = =

**Product of the zeroes**

= = =

Example: Find the zeroes of the quadratic polynomial 9x^{2}– 5 and verify the relation between the zeroes and its coefficients.

Solution: We have,

9x^{2} – 5

= (3x)^{2} – (√5)^{2}

= (3x – √5) (3x +√5)

So, the value of 9x^{2} – 5 is 0, when

3x –√5 = 0 or 3x +√5 = 0

i.e., when x = or x = .

**Sum of the zeroes**

=+ = 0 = =

**Product of the zeroes**

= ×= =

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