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If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and c all have the same sign. Is the following statement ‘True’ or ‘False’? Justify your answer

Solution:

Given, the quadratic polynomial is ax² + bx + c.

The zeros of the polynomial are both positive.

We have to determine whether a, b and c all have the same sign.

We know that, if 𝛼 and ꞵ are the zeroes of a polynomial ax² + bx + c, then

Sum of the roots is 𝛼 + ꞵ = -b/a

Product of the roots is 𝛼ꞵ = c/a

Sum of the roots = 𝛼 + ꞵ

𝛼 + ꞵ = -b/a < 0

Sum of the roots is negative.

So, one of the zeros is negative.

a and b have different signs.

Therefore, a, b and c all cannot have the same sign.

✦ Try This: Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial rx² + sx + u are both positive, then r, s and u all have the same sign

Given, the quadratic polynomial is rx² + sx + u.

The zeros of the polynomial are both positive.

We have to determine whether r, s and u all have the same sign.

We know that, if 𝛼 and ꞵ are the zeroes of a polynomial ax² + bx + c, then

Sum of the roots is 𝛼 + ꞵ = -b/a

Product of the roots is 𝛼ꞵ = c/a

Here, 𝛼 = -𝛼, ꞵ = -ꞵ, a = r, b = s, c = u

Sum of the roots = 𝛼 + ꞵ

-b/a = -s/r

𝛼 + ꞵ = -s/r < 0

Sum of the roots is negative.

So, one of the zeros is negative.

r and s have different signs.

Therefore, r, s and u all cannot have the same sign. The statement is false.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.2 Problem 2 (i)

If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and c all have the same sign. Is the following statement ‘True’ or ‘False’? Justify your answer

Summary:

If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and call cannot have the same sign. The statement is false

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