Question

find the sum and the product of zeros of the polynomial 2 x square minus x minus 1 and verify the relation with the coefficient​

Answer 1

Answer:

1/2 ; -1/2.

Step-by-step explanation:

There are two methods to solve this,

Method - 1:

2x² - x - 1 = 0

2x² - 2x + x - 1 = 0

2x(x - 1) +1(x - 1) = 0

(2x + 1)(x - 1) = 0

2x + 1 = 0

2x = -1

x = -1/2.

x - 1 = 0

x = 1.

Sum of the zeroes = -1/2 + 1

= (-1+2)/2

Sum of the roots = 1/2

Product = (-1/2)*(1)

product of the roots = -1/2.

Method - 2:

Let a quadratic equation be ax² + bx + c = 0

We know that, if p and q are the roots of the

quadratic equation then,

Sum of the roots (p+ q) = - b/a

Product of the roots pq = c/a

Hence from the above equation i.e., 2x² - x - 1 = 0

Here , a = 2 ; b = -1 ; c = -1

Sum of the roots = - b/a

= -(-1/2)

Sum of the roots = 1/2

Product of the roots = c/a

Product of the roots = -1/2

Answer 2

❏ Question

find the sum and the product of zeros of the polynomial 2 x square minus x minus 1 and verify the relation with the coefficients

❏ Solution

Given:-

• equation 2x² - x - 1 = 0 .

Find:-

• Sum of zeroes .
• Product of zeroes .
• Verify the relation with the coefficient .

❏Explanation

Case(1):-

First calculate zeroes

➠ 2x² - x - 1 =0

➠ 2x² - 2x + x - 1 = 0

➠ 2x(x-1)+1(x-1) = 0

➠ ( 2x+1)(x-1)=0

➠ (2x+1)=0 Or, (x-1)=0

➠ x = -1/2 Or, x = 1

Zeroes of given equation -1/2 , 1

★ Sum of zeroes = (-1/2)+1

★ Sum of zeroes = (-1+2)/2

★ Sum of zeroes = 1/2

And,

★ Product of zeroes = (-1/2)×(1)

★ Product of zeroes = -1/2

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❏ Verification

We know

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

➠ Sum of zeroes = -(-1)/(2)

➠ Sum of zeroes = 1/2

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★Product of zeroes = (Constant part )/(Coefficient of x²)

➠ Product of zeroes = -1/2

Here,condition are satisfied

Hence proved.

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