Question

How do you simplify `2(sqrt(5x) - 3)^2`?

Answer 1

`10x-12sqrt(5x)+18`

Explanation:

We can rewrite the expression to specifically show all the terms:

`2(sqrt(5x)-3)^2=2(sqrt(5x)-3)(sqrt(5x)-3)`

Let's set aside the leading 2 for a minute and work with the two bracketed terms. We'll use FOIL to handle it:

FOIL

• `color(red)(F)` - First terms - `(color(red)(a)+b)(color(red)(c)+d)`
• `color(brown)(O)` - Outside terms - `(color(brown)(a)+b)(c+color(brown)d)`
• `color(blue)(I)` - Inside terms - `(a+color(blue)b)(color(blue)(c)+d)`
• `color(green)(L)` - Last terms - `(a+color(green)b)(c+color(green)d)`

This gives us:

• `color(red)(F)=>sqrt(5x)sqrt(5x)=5x`
• `color(brown)(O)=>sqrt(5x)(-3)=-3sqrt(5x)`
• `color(blue)(I)=>(-3)(sqrt(5x))=-3sqrt(5x)`
• `color(green)(L)=>(-3)(-3)=9`

`5x-3sqrt(5x)-3sqrt(5x)+9=5x-6sqrt(5x)+9`

And so `2(sqrt(5x)-3)^2=2(5x-6sqrt(5x)+9)`

We can now distribute the 2 through the bracket:

`2(5x-6sqrt(5x)+9)=color(blue)(ul(bar(abs(color(black)(10x-12sqrt(5x)+18))))`