Question

Find the zeroes of the quadratic polynomial 3x*x+5x-2 and verify the relationship between zeroes and the coefficient

Answer 1

Question :-

Find the zeros of the quadratic polynomial 3x²+5x-2 and verify the relationship between zeroes and the coefficient.

Answer :-

→ -2 and 1/3 are zeros of the given quadratic equation .

To Find :-

→ Zeros of given quadratic equation .

Explanation :-

We have :-

→ 3x² + 5x - 2 = 0

split the middle term .

→ 3x² + 6x - x - 2 = 0

→ 3x( x + 2) -1( x + 2) = 0

→ (x + 2)(3x - 1) = 0

• Case (1) if -

→ x + 2 = 0

→ x = -2

• Case (2) if -

→ 3x - 1 = 0

→ 3x = 1

→ x = 1/3

Hence ,

→ Zeros are 1/3 and -2

• Coefficient of x² = 3

• Coefficient of x = 5

Verification :-

If x = -2

→ 3(-2)² + 5× (-2) -2 = 0

→ 12 - 10 - 2 = 0

→ 12 - 12 = 0

→ 0 = 0

If x = 1/3

→3 (1/3)² + 5/3 - 2 = 0

→ (1 + 5 - 6)/3 = 0

→ (6-6)/3 = 0

→ 0 = 0

Hence verified .

Relationship between zeros and coefficients :-

(•) Sum of zeros = 1/3 -2 = -5/3

And in quadratic equation

sum of zeros = -b/a = -5/3

(•) Product of zeros = -2/3

and in quadratic equation product of zeros = c/a = -2/3

Hence verified .

Answer 2

Solution :

$\rule{100}2$

First of all, factorise 3x² + 5x - 2 by splitting the middle term.

$\sf = 3x^2 + 5x - 2$

$\sf = 3x2 + ( 6 - 1 )x - 2$

$\sf = 3x^2 + 6x - x - 2$

$\sf = 3x ( x + 2 ) - 1 ( x + 2 )$

$\sf = ( 3x - 1 ) ( x + 2 )$

Now, to find the zeroes -

3x - 1 = 0

⇒ 3x = 1

⇒ x = 1/3  

∴ α = 1/3

For x + 2 :-

x + 2 = 0

⇒ x = -2

∴ β = - 2

Finally, verification :-

Here, a = 3 ; b = 5 ; c = - 2

We know that,

α + β = - b/a

$\sf\implies\dfrac{1}{3} + ( - 2 ) = \dfrac{ - ( 5)}{3}$

$\sf\implies \dfrac{ 1 - 6}{3}= \dfrac{-5}{3}$

∴ $\sf \dfrac{-5}{3}=\dfrac{-5}{3}$

L.H.S = R.HS

Also,

αβ = c/a

$\sf\implies \dfrac{1}{3} \times \dfrac{-2}{1} = \dfrac{-2}{3}$

$\sf\implies\dfrac{-2}{3} =\dfrac{-2}{3}$

∴ L.H.S = R.H.S

Hence verified!

$\rule{200}2$