Question

Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the cofficients i) 3x^2-x-4

Answer 1

• a = 3

• b = -1

• c = -4

We have,

3x² - x - 4

⇒ 3x² + 3x - 4x - 4

⇒ 3x ( x + 1 ) - 4 ( x + 1 )

⇒ ( x + 1 ) ( 3x - 4 )

So,to find zeroes of polynomial:

3x² - x - 4 will be 0, hence ( x + 1 ) = 0 and ( 3x - 4 ) = 0

⇒ x = -1 and x = 4/3

Therefore,

Sum of the zeroes : α + β = -b/a

⇒ - 1 + 4/3 = - ( -1 )/3

⇒ -3 + 4 / 3 = 1/3

⇒ 1/3 = 1/3

Product of the zeroes: αβ = c/a

⇒ ( - 1 ) × 4/3 = -4/3

⇒ -4/3 = -4/3

Verified.

Answer 2

$\huge\mathfrak\red{Answer:}$

Given:

• We have been given a quadratic equation 3x^2-x-4.

To Find:

• We need to find the zeroes of this equation and also verify the relationship between the zeroes and coefficients.

Solution:

The given quadratic polymial is 3x^2-x-4.

We can find the zeroes of this polynomial by the method of splitting the middle term.

We need to find two such numbers whose sum is -1 and product is -12.

Two such numbers are +3 and -4, so we have

$\sf{ {3x}^{2} + 3x - 4x - 4 = 0}$

$\implies\sf{3x(x + 1) - 4(x + 1) = 0}$

$\implies\sf{(x + 1)(3x - 4) = 0}$

Either (x + 1) = 0 or (3x - 4) = 0.

When x + 1 = 0

=> x = -1

When 3x - 4 = 0

=> 3x = 4

=> x = 4/3

Therefore, two zeroes of this polynomial are -1 and 4/3.

Now, we need to verify the relationship between zeroes and coefficients.

Sum of zeroes = A + B

= (-1) + (4/3)

= 1/3= -b/a

Product of zeroes = AB

= (-1) × (4/3)

= -4/3 = c/a

A + B = -b/a and AB = c/a.

Hence, relationship between zeroes and coefficients is verified!!