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UPSC NDA II 2015 Mathematics Paper

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Question : 12 of 120

Marks: +1, -0

The number of real roots of the equation

x^{2} - 3 | x | + 2 = 0

is

4
3
2
1

Solution:

Concept:

The key property is

x^{2} = |x|^{2}

, which allows rewriting the equation in terms of

|x|

.

Explanation:

Start with

x^{2} - 3|x| + 2 = 0

.

Replace

x^{2}

with

|x|^{2}

, giving

|x|^{2} - 3|x| + 2 = 0

.

Factor the quadratic in

|x|

:

|x|^{2} - 2|x| - |x| + 2 = 0

.

Group terms:

|x|(|x| - 2) - 1(|x| - 2) = 0

, so

(|x| - 2)(|x| - 1) = 0

.

Thus

|x| = 2

or

|x| = 1

.

From

|x| = 2

, we get

x = \pm 2

. From

|x| = 1

, we get

x = \pm 1

.

Hence there are four real roots:

-2

,

-1

,

1

, and

2

.

Answer:

A. 4

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