Question

Sviatoslav solved the quadratic equation x^2-x-1=0 by completing the square. In the process, he came up with the equivalent equation (x+a)^2 = b, where a and b are constants. What is b?

Answer 1

`x=(1+\sqrt5)/(2)` and
`x=(1-\sqrt5)/(2)`

and b=
`(5)/(4)`

Explanation:

We are given that a quadratic equation

`x^2-x-1=0`

We have to solve the equation by completing square and find the value of b.

`(x)^2-2* x* (1)/(2)+((1)/(2))^2-((1)/(2))^2-1=0`

`(x-(1)/(2))^2-(1)/(4)-1=0`

`(x-(1)/(2))^2-(1-4)/(4)=0`

`(x-(1)/(2))-(5)/(4)=0`

`(x-(1)/(2))^2-((\sqrt5)/(2))^2=0`

`(x-(1)/(2))^2=((\sqrt5)/(2))^2`

`x-(1)/(2)=(\sqrt5)/(2)` and
`x-(1)/(2)=-(\sqrt5)/(2)`

`x=(\sqrt5)/(2)+(1)/(2)=(1+\sqrt5)/(2)`

And
`x=(1)/(2)-(\sqrt5)/(2)=(1-\sqrt5)/(2)`

Therefore,
`x=(1+\sqrt5)/(2)` and
`x=(1-\sqrt5)/(2)`

and b=
`(5)/(4)`