Question

find the zeroes of the quadratic polynomial 3x2-x-4 and verify the relationship between the zeroes and coefficient.​

Answer 1

Solution

Given :-

• Equation, 3x² - x - 4 = 0

Find :-

• Its roots

Explanation

Let,

P & Q be roots of this Equation.

Using Formula

★ Sum of roots = - ( Coefficient of x)/(coefficient of x²)

★ product of roots = (constant part)/(coefficient of x²)

So, keep above Value,

==> sum of roots = -(-1)/3

==> P + Q = 1/3____________(1)

And,

==> Product of roots = -4/3

==> P.Q = -4/3

==> P = -4/(3Q)____________(2)

keep in equ(1)

==> -4/(3Q) + Q = 1/3

==> (-4 + 3Q² ) × 3 = 3Q

==> 9Q² - 3Q - 12 = 0

==> 3Q² - Q - 4 = 0

==> 3Q² - 4Q + 3Q - 4 = 0

==> Q(3Q - 4) + 1(3Q - 4) = 0

==> (3Q - 4)(Q + 1)= 0

==> 3Q - 4 = 0 Or, Q + 1 = 0

==> Q = 4/3 Or, Q = -1

keep value of Equation (1)

When,

• Q = 4/3

==> P + 4/3 = 1/3

==> P = 1/3 - 4/3

==> P = (1- 4)/3

==> P = -3/3

==> P = -1

When,

• Q = -1,

==> P - 1 = 1/3

==> P = 1/3 + 1

==> P = (1 + 3)/3

==> P = 4/3

Since

• Roots will be of this Equation, 4/3 , -1 Or, -1 , 4/3.

______________________________

Answer Verification

★Sum of roots = 1/3

==> 4/3 - 1 = 1/3

==> ( 4 - 3)/3 = 1/3

==> 1/3 = 1/3

L.H.S. = R.H.S.

That's Proved.

__________________

Answer 2

Corrected Question :-

• Find the zeroes of the quadratic polynomial 3x²-x-4 and verify the relationship between the zeroes and coefficient.

Answer :-

Given :-

• polynomial :=> 3x²-x-4.

To find :-

• Find the zeroes of the polynomial.
• Verify the relationship between the zeroes and coefficient.

Solution :-

⇒3x²-x-4 = 0

→splitting the middle term

⇒ 3x²+3x-4x-4 = 0

⇒ 3x(x+1)-4(x+1) = 0

⇒ x+1 = 0 ,  3x-4 =0

⇒ x = 0-1  ,  3x = 0+4

⇒ x = -1  , x = 4/3

→ Two zeroes of this polynomial are :⇒ -1 & 4/3

→ Let α= -1  β= 4/3

→Verification of relationship between the zeroes and coefficient.

⇒sum of zeroes = -b/a

⇒ α+β = -b/a

⇒ -1 + 4/3 = -(-1)/3

⇒ -3+4/3 = 1/3

⇒ 1/3 = 1/3

⇒product of zeroes = c/a

⇒ α×β = -4/3

⇒ -1 × 4/3 = -4/3

⇒ -4/3 = -4/3

→ Hence verified

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