Question

x²+x+1=(x+1/2)² + (√3/2)² how is any formula to convert it into that​

Answer 1

☆ The method is called Completing Squares Method

$\tt \longrightarrow \: {x}^{2} + x + 1$

☆ On adding and Subtracting square of 1/2 the coefficient of x, we get

$\tt \longrightarrow \: {x }^{2} + x + \dfrac{1}{4} - \dfrac{1}{4} + 1$

$\tt \longrightarrow \: ( {x}^{2} + x + \dfrac{1}{4} ) + \dfrac{4 - 1}{4}$

$\tt \longrightarrow \: { \bigg(x + \dfrac{1}{2} \bigg)}^{2} + \dfrac{3}{4}$

$\tt \longrightarrow \: { \bigg(x + \dfrac{1}{2} \bigg)}^{2} + { \bigg(\dfrac{ \sqrt{3} }{2} \bigg)}^{2}$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$