Question

Find the zeroes of the quadratic polynomial given below Find the sum and product of the zeroes and verify relationship to the coefficient of terms in the polynomial p(x)=x²-x-6

Answer 1

$\large\bf\underline{Given:-}$

• p(x) =x²-x-6

$\large\bf\underline {To \: find:-}$

• zeroes of given polynomial.
• relationship between the zeroes and coefficients.

$\huge\bf\underline{Solution:-}$

• p(x) = x²-x-6

⠀⠀⠀➝ x²-x-6

⠀⠀⠀➝ x²-3x +2x -6

⠀⠀⠀➝ x(x-3) +2(x-3)

⠀⠀⠀➝ (x-3)(x+2)

• x = 3 or x = -2

Let α and β are the zeroes of the given polynomial.

• α = 3
• β = -2

p(x) = x²-x-6

a = 1

b = -1

c = -6

sum of zeroes = - b/a

α + β = -b/a

➝ 3+(-2) = -(-1)/1

➝ 3 -2 = 1

➝ 1 = 1

product of zeroes = c/a

➝ αβ = c/a

➝ 3 × -2 = -6/1

➝ 3 × -2 = -6

➝ -6 = -6

LHS = RHS

hence relationship is verified

Answer 2

GIVEN:

• Quadratic polynomial

TO FIND:

• Relationship b/w the roots

• roots of polynomial

SOLUTION:

Let

• roots be α , β

p(x) = x² - x - 6

Putting p(x) = 0

x² - x - 6 = 0

Finding roots by quadratic formula

x = -b±√b² - 4ac/2a

• b : coefficient of x = -1

• a : coefficient of x² = 1

• c : constantly term = 6

Substituting the values we have

→ x = - ( - 1)±√(-1)² - 4(1)(-6)/2(1)

→ x = 1±√1 + 24/2

→ x = 1±√25/2

→ x = 1+5/2 or 1 - 5/2

α = 6/2 = 3

β = -4/2 = -2

Now finding relationship

α + β = -b/a

3 - 2 = -(-1)/1

1 = 1

____________________

αβ = c/a

3(-2) = -6/1

-6 = -6

Hence relationship is verified