Question

13. Find the quadratic polynomial whose zeros are 2 and -6. Verify the
relation between the coefficients and the zeros of the polynomial.​

Answer 1

Solution

Given :-

• Zeroes of any polynomial 2 & -6 .

Find :-

• Equation of polynomial ,
• relation between the coefficients and the zeros of the polynomial.

Explanation

Let, here

• first zeroes (p) = 2
• Second zeroes (q) = -6

So,

★ Sum of zeroes (p + q) = 2 + (-6)

==> Sum of zeroes (p + q) = -4.-----(1)

And,

★ Product of zeroes p . q = 2 * (-6)

==> Product of zeroes (p . q) = -12 -------(2)

________________________

Equation ,formula

★ x² - ( sum of zeroes) x + product of zeroes = 0

Keep all above value,

==> x² - (-4)x + (-12) = 0

==> x² + 4x - 12 = 0

Hence, Required equation

• x² + 4x - 12 = 0

________________

Verification of zeroes & coefficient of polynomial

Important Formula

★ Sum of zeroes = -b/a

★ product of zeroes = c/a

Where,

• a = Coefficient of x²
• b = Coefficient of x
• c = Constant part

So,

==> Sum of zeroes = -4/1

==> p + q = -4 -------------(3)

And,

==> product of zeroes = -12/1

==> p . q = -12 --------------(4)

Here, equ(1) , equ(3) & equ(2) , equ(4) are same

Hence proved.

___________________

Answer 2

Given that:

• Roots of a polynomial is 2 and -6.

To Find:

• The quadratic equation and verify the relationship between roots and coefficients.

Formula Used:

• For a quadratic equation f(x) having root as p and q, we have

f(x) = (x-p)(x-q) = 0

SOLUTION:

Let the quadratic equation be f(x).

So, using the above formula, we have

f(x) = (x-2)(x+6)

f(x) = x^2-2x+6x-12

f(x) = x^2+4x-12 = 0

So, the quadratic equation is x^2+4x-12 = 0.

Verification:

• Sum of roots = -b/a
• Product of roots = c/a

Now, ATQ, we have

b = 4 ,c = -12 and a = 1.

So, 2+(-6) = -4

-4 = -4...(1)

Also, (2)*(-6) = -12/1

-12 = -12...(2)

From eq(1) and eq(2), we get

LHS = RHS.

Hence, verified !

Hope this helps