Question

If x2 = 18, then check whether x is rational or irrational

Answer 1

Heya !!!






✓18 = 2 × 3 × 3


X² = 18




X = ✓18





X = ✓3 × 3 × 2






X = 3✓2 which is irrational number.





Hence,



X is irrational number.






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Answer 2

If x² = 18 then x is an irrational number

Given :

x² = 18

To find :

x is a rational number or an irrational number

Solution :

Step 1 of 4 :

Write down the given equation

Here the given equation is

x² = 18

Step 2 of 4 :

Find the value of x

$\displaystyle \sf{ {x}^{2} = 18 }$

$\displaystyle \sf{ \implies x = \pm \: \sqrt{18} }$

$\displaystyle \sf{ \implies x = \pm \:3 \sqrt{2} }$

Step 3 of 4 :

Prove that √2 is an irrational number

Let us assume that √2 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

⇒ √2 = p/q { where p and q are co- prime}

⇒ √2q = p

Now, by squaring both the side

we get,

(√2q)² = p²

⇒ 2q² = p² - - - - - - (1)

Now if 2 is the factor of p²

Then , 2 is also a factor of p - - - - - - (2)

⇒ Let p = 2m { where m is any integer }

squaring both sides

p² = (2m)²

p² = 4m²

Putting the value of p² in Equation 1 we get

2q² = p²

⇒ 2q² = 4m²

⇒ q² = 2m²

Now if 2 is factor of q²

Then , 2 is also factor of q

Since

2 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Hence √2 is an irrational number

Step 4 of 4 :

Find x is a rational number or an irrational number

Since √2 is an irrational number

∴ 3√2 is an irrational number and - 3√2 is also an irrational number

∴ x is an irrational number

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