IPMAT Indore 2022 - QA (MCQ) Q16 Explanation

Explanation:

(a)

𝑥2 + |𝑥+4| + |𝑥−4| − 35 = 0

Case I: if x ≥ 4

Then, 𝑥^2+(𝑥+4)+(𝑥−4)−35=0

𝑥^2 + 2𝑥 − 35 = 0

𝑥^2 + 7𝑥 − 5𝑥 − 35 = 0

𝑥(𝑥+7) −5(𝑥+7) = 0

(𝑥−5)(𝑥+7) =

𝑥 = 5 or 𝑥 = −7

Only x= 5 satisfies the assumption.

Case II: If 𝑥<−4.

Then 𝑥^2 − (𝑥+4) − (𝑥−4) − 35 = 0.

𝑥^2 − 2𝑥 − 35 = 0

𝑥^2 + 5𝑎 − 7𝑥 − 35 = 0

(𝑥+5) −7 (𝑥+5) = 0

(𝑥−7)(𝑥+5) = 0

X = 7 or x = -5

Only x = -5 satisfies the assumptions.

Case III: If −4 < 𝑥 ≤4

Then, 𝑥^2 + (𝑥+4) − (𝑥−4) − 35 = 0

𝑥^2 + 𝑥 + 4 − 𝑥 + 4 − 35 = 0

𝑥^2 − 27 = 0

𝑥 = +√27 or 𝑥 = -√27 

None of these values of ‘x’ satisfies the assumption.

∴ only 2 roots i.e. 5 and –5 will be there

Their sum = 52 + (−5)^2 = 25 + 25 = 50