Question

How do you solve `x^2-14x+5=0` by completing the square?

Answer 1

`x=7-2sqrt11` and `x=7+2sqrt11`

Explanation:

Half the second term of `x` put it in brackets and square it.

`x^2-14x+5-> (x-7)^2+5`

Take away the squared number of the number inside the bracket.

`7^2=49` `therefore` `-49`

`-> (x-7)^2-49+5`

Simplify:

`(x-7)^2-44`

Solving:

Set equal to `0`:

`(x-7)^2-44=0`

Add `44`:

`(x-7)^2=44`

Square root:

`x-7=pmsqrt44`

Add `7`:

`x=7pmsqrt44`

Simplifying:

`sqrt44=sqrt4 xx sqrt11`

`sqrt4=2`

`sqrt11=sqrt11`, It cannot be simplified as it is a prime number.

`therefore` `x=7pmsqrt44 -> x=7pm2sqrt11`

Therefore both answers are:

`x=7-2sqrt11`

and...

`x=7+2sqrt11`