Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
(i) xsquare -2x-8

Answer 1

Given :

Quadratic polynomial

• x² - 2 x - 8

To find :

Zeroes of the given Quadratic polynomial and verifying the relationship between zeroes and coefficients.

Knowledge required :

For a given quadratic polynomial

ax² + bx + c

• sum of zeroes = -b/a
• product of zeroes = c/a

Solution :

Finding the zeroes of given polynomial by splitting the middle term

→ x² - 2 x - 8 = 0

→ x² - 4 x + 2 x - 8 = 0

→ x ( x - 4 ) + 2 ( x - 4 ) = 0

→ ( x + 2 ) ( x - 4 ) = 0

→ x = -2  or x = 4

therefore,

two zeroes of given quadratic polynomial are -2 and 4.

Verifying the relationship between zeroes and coefficients

Comparing x² - 2 x - 8 with ax² + bx + c

we will get,

• a = 1
• b = -2
• c = -8

so,

→  sum of zeroes = -b/a

→ -2 + 4 = - ( -2) / 1

→ 2 = 2

verified .

and

→ product of zeroes = c/a

→ (-2) (4) = (-8) / 1

→ - 8 = - 8

verified .

Answer 2

$\sf{\underline{Question :-}}$

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.

(i) x²-2x-8

$\sf{\underline{Solution :-}}$

† Using splitting method †

→ x² - 2x - 8 = 0

→ x² - ( 4 - 2 )x - 8 = 0

→ x² - 4x + 2x - 8 = 0

→ x( x - 4) + 2 ( x - 4 ) = 0

→ ( x - 4 ) ( x + 2 ) = 0

→ x - 4 = 0

→ x = 4

★ either

→ x + 2 = 0

→ x = -2

★ Now,

†Verification between the zeroes and coefficients†

• we know

General equation

★ ax² + bx + c

Now,

comparing with general equation x²-2x-8

→ a = 1 or x

→ b = -2

→ c = -8

We know

★ Sum of zeroes = -b/a

→ 4 + (-2) = -2/1

→ 4 -2 = -2/1

→ -2 × 1 = -2

→ -2 = -2

→ 2 = 2 ( verified )

Now,

★ product of zeroes = c/a

→ 4 × ( -2 ) = -8/1

→ -8 × 1 = - 8

→ - 8 = 8

→ 8 = 8 ( verified )