Question

Solve each equation by completing the square. X^2+10x=17

Answer 1

Completing squares

Before attempting to complete squares, let's recall the following identity

`(a+b)^2=a^2+2ab+b^2`

the expression at the right side can be converted to the square of a binomial, provided we have the terms completed as shown

We have the equation:

`x^2+10x=17`

note the left side has TWO of the terms required for the square of a binomial. we only need the final number. but what number should we add?

the first term is the square of a, in this case, it's x

the second term has 10x and it should be 2ab, if we already know a=x, then

2ab=10x, then

b=10x/(2x)=5

now we know a=x and b=5, we only need to have b^2=25

that is exactly the number to add on both sides of the equation

`x^2+10x+25=17+25=42`

now we factor the left side:

`(x+5)^2=42`

taking the square root, recall the square root can