Question

Find the values of a and b: √3 − 2 / √3 +2 = a + b √3

Answer 1

Step-by-step explanation:

$\longmapsto \rm {\dfrac{ \sqrt{3} - 2}{ \sqrt{3} + 2 } = a + b \sqrt{3} }$

Step 1 : Rationalising the denominator in L. H. S. In order to rationalise the denominator, we multiply the rationalising factor of denominator with both the numerator and the denominator. Here, rationalising factor of (√3 + 2) is (√3 - 2).

$\longmapsto \rm { \dfrac{ \sqrt{3} - 2 }{ \sqrt{3} + 2 } \times \dfrac{ \sqrt{3} - 2 }{ \sqrt{3} - 2 } = a + b \sqrt{3} }$

Step 2 : Multiplying (√3 - 2) with both the numerator and the denominator.

$\longmapsto \rm {\dfrac{ {( \sqrt{3} - 2) }^{2} }{ (\sqrt{3} + 2)( \sqrt{3} - 2) } = a + b \sqrt{3} }$

Step 3 : Rearranging the terms in the form of identities.

$\longmapsto \rm {\dfrac{ ( \sqrt{3} ) ^{2} + (2)^{2} - 2( \sqrt{3} )(2)}{ { (\sqrt{3}) }^{2} - {(2)}^{2} } = a + b \sqrt{3} }$

Step 4 : By using the identities, solving further.

• (a - b)² = a² + b² - 2ab
• (a + b)(a - b) = a² - b²

$\longmapsto \rm {\dfrac{ 3 + 2 - 4\sqrt{3} }{ 3 - 4 } = a + b \sqrt{3} }$

Step 5 : Simplifying further.

$\longmapsto \rm {\dfrac{ 5 - 4\sqrt{3} }{ - 1 } = a + b \sqrt{3} }$

Step 4 : Performing addition and subtraction in the numerator and the denominator.

$\longmapsto \rm {\dfrac{ 5 - 4\sqrt{3} }{ - 1 } \times \dfrac{ - 1}{ - 1} = a + b \sqrt{3} }$

Step 5 : Making the denominator positive by multiplying -1 with both the numerator and the denominator.

$\longmapsto \rm {\dfrac{ - 1(5 - 4\sqrt{3}) }{ ( - 1) ^{2} } = a + b \sqrt{3} }$

Step 6 : Multiplying -1 with all the terms in the bracket.

$\longmapsto \rm {\dfrac{ - 5 + 4\sqrt{3}}{1 } = a + b \sqrt{3} }$

$\longmapsto \rm { - 5 + 4\sqrt{3}= a + b \sqrt{3} }$

Step 7 : Comparing L. H. S and R. H. S.

$\longmapsto \rm { \red{ - 5 } + \red{4}\sqrt{3} \Leftrightarrow \red{ a }+ \red{b} \sqrt{3} }$

Step 8 : After comparison, we got the value of a and b that is -5 and 4.

Therefore, value of a is -5 and b is 4.