Exercise 2.2, 1

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² – 2x – 8

Solution:

Given Polynomial:

p(x) = x² − 2x − 8

Finding the Zeroes

We solve the equation:

x² − 2x − 8 = 0

Using the factorization method, we find two numbers whose product is -8 and sum is -2. These numbers are -4 and 2.

Rewriting the equation:

x² − 4x + 2x − 8 = 0

Grouping the terms:

x(x − 4) + 2(x − 4) = 0

Taking out the common factor:

(x − 4)(x + 2) = 0

Setting each factor to zero:

x − 4 = 0 ⇒ x = 4

$$x+2=0\Rightarrow x=-2$$

Thus, the zeroes are 4 and -2.

Verification of the Relationship

For a quadratic equation of the form:

ax² + bx + c

• Sum of zeroes formula:

  Sum of zeroes =−coefficient of x/coefficient of x²
  α + β = −a/b​

• Product of zeroes formula:

  Product of zeroes = constant term/coefficient of x²
  α × β = ac​

For x²−2x−8 we have:
$a=1$, b = -2, c=−8.

Checking the Sum of Zeroes:

α + β = 4 + (−2) = 2

−1/2​ = 2

Since L.H.S = R.H.S, the sum is verified. ✅

Checking the Product of Zeroes:

α×β=4×(−2)=−8

-8/1 ​= −8

Since L.H.S = R.H.S, the product is verified. ✅

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(ii) 4s² – 4s + 1

Solution:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(iii) 6x² – 3 – 7x

Solution:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(iv) 4u² + 8u

Solution:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(v) t2 – 15

Solution:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(vi) 3x² – x – 4

Solution: