Question

x=1/3+root 5 find the value of 16x^2 -14

Answer 1

AnSwEr

${\tt{\red{\underline{\large{Given}}}}}$

• $\tt\dfrac{1}{3+\sqrt{5}}$

${\sf{\green{\underline{\large{To\:find}}}}}$

• value of 16x²-14

${\sf{\pink{\underline{\Large{Explanation}}}}}$

First Step

we have to rationalize the given y

$\tt\dfrac{1}{3+\sqrt{5}}\times\dfrac{3-\sqrt{5}}{3-\sqrt{5}}$

$\implies\tt\dfrac{1}{3+\sqrt{5}}\times\dfrac{3-\sqrt{5}}{3-\sqrt{5}}$

$\implies\tt\dfrac{3-\sqrt{5}}{4}$

¶utting the value

$\implies\tt{X=}\dfrac{3-\sqrt{5}}{4}$

Then

16x²-14

$\implies\tt\dfrac{16(3-\sqrt{5})^2}{4}-14$

$\implies\tt\dfrac{16(3-\sqrt{5}^2)}{4^2}-14$

$\implies\tt\dfrac{16(9+5-6\sqrt{5})}{16}-14$

$\implies\tt\:X=14-6\sqrt{5}-14$

$\implies\tt\:X=-6\sqrt{5}$

Answer 2

Answer:

16x² - 14 = - 6√5

Solution:

• Given : x = 1/(3 + √5)
• To find : 16x² - 14 = ?

We have ;

x = 1/(3 + √5)

Now,

Rationalising the denominator of the term in RHS , we have ;

=> x = (3 - √5) / (3 + √5)(3 - √5)

=> x = (3 - √5) / (3² - √5²)

=> x = (3 - √5) / (9 - 5)

=> x = (3 - √5) / 4

=> 4x = 3 - √5

Now,

Squaring both the sides , we get ;

=> (4x)² = (3 - √5)²

=> 4²x² = 3² - 2•3•√5 + √5²

=> 16x² = 9 - 6√5 + 5

=> 16x² = 14 - 6√5

=> 16x² - 14 = - 6√5

Hence,

16x² - 14 = - 6√5 .