Question

The number of real roots of the equation ${x^2} - 3\left| x \right| + 2 = 0$is
A.$4$
B.$3$
C.$2$
D.$1$

Answer 1

Hint: Here we will take the given expression and convert the modulus in the form of a simple equation and use the split method to get the factors of the equation. Then identify the roots of the equations which are real and then find the number of real roots accordingly.

Complete step-by-step answer:
Take the given expression: ${x^2} - 3\left| x \right| + 2 = 0$
Here modulus is given, it means the value can be plus or minus. Therefore, equation becomes –
${x^2} - 3( \pm x) + 2 = 0$
Here, we have two cases –
${x^2} - 3( + x) + 2 = 0$ and b) ${x^2} - 3( - x) + 2 = 0$
Take first equation –
${x^2} - 3( + x) + 2 = 0$
Simplify the above expression finding the product of negative and the positive term. Product of minus with plus gives minus.
${x^2} - 3x + 2 = 0$
Split the middle term –
${x^2} - 2x - x + 2 = 0$
Make the pair of first two terms and last two terms –
$\underline {{x^2} - 2x} - \underline {x + 2} = 0$
Take out common multiple from the paired terms –
$x(x - 2) - 1(x - 2) = 0$
Take common term common –
$(x - 2)(x - 1) = 0$
$x - 2 = 0$or $x - 1 = 0$
Make the required term the subject, when you move any term from one side to the opposite side then the sign of the term also changes. Positive term becomes negative and vice-versa.
$ \Rightarrow x = 1,2$ …. (A)

Now, take the second equation –
${x^2} - 3( - x) + 2 = 0$
Product of minus with minus gives plus.
${x^2} + 3x + 2 = 0$
Split the middle term –
${x^2} + 2x + x + 2 = 0$
Make the pair of first two terms and last two terms –
$\underline {{x^2} + 2x} + \underline {x + 2} = 0$
Take out common multiple from the paired terms –
$x(x + 2) + 1(x + 2) = 0$
Take common term common –
$(x + 2)(x + 1) = 0$
$x + 2 = 0$or $x + 1 = 0$
Make the required term the subject, when you move any term from one side to the opposite side then the sign of the term also changes. Positive term becomes negative and vice-versa.
$ \Rightarrow x = - 1, - 2$ …. (B)
From equations (A) and (B) –
The real roots of the given equations are - $1,2, - 1, - 2$
Hence, from the given multiple choices – the option A is the correct answer.
So, the correct answer is “Option A”.

Note: Reals numbers can be defined as the numbers which include rational as well as irrational numbers. Be careful about the sign convention while simplifying the equations. Remember the product of two same signs gives the positive term while the product of terms with two different signs, then the resultant term is negative.