A day full of math games & activities. Find one near you.

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

Solution:

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

The zeros of the polynomial are both negative.

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

Here, 𝛼 = -𝛼

ꞵ = -ꞵ

Sum of the roots = -𝛼 - ꞵ = -(𝛼 + ꞵ)

-(𝛼 + ꞵ) = -b/a

𝛼 + ꞵ = b/a > 0

Product of the roots = (-𝛼)(-ꞵ) = 𝛼ꞵ

𝛼ꞵ = c/a > 0

Therefore, a, b and c all 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 negative, 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 negative.

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 = -(𝛼 + ꞵ)

-b/a = -s/r

-(𝛼 + ꞵ) = -s/r

𝛼 + ꞵ = s/r > 0

Product of the roots = (-𝛼)(-ꞵ) = 𝛼ꞵ

c/a = u/r

𝛼ꞵ = u/r > 0

Therefore, r, s and u all have the same sign

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

NCERT Exemplar Class 10 Maths Exercise 2.2 Solved Problem 2

Summary:

If the zeroes of a quadratic polynomial ax² + bx + c are both negative, then a, b and c all have the same sign. The statement is true

☛ Related Questions: