Question

We want to factor the following expression: (x 3)^2 14(x 3) 49(x 3) 2 14(x 3) 49(, x, plus, 3, ), squared, plus, 14, (, x, plus, 3, ), plus, 49 We can factor the expression as (U V)^2(U V) 2 (, U, plus, V, ), squared where UUU and VVV are either constant integers or single-variable expressions. 1) What are UUU and VVV

Answer 1

`U = x`

`V=10`

Explanation:

Given

`(x +3)^2 + 14(x + 3) + 49 = (U + V)^2`

Required

Find U and V

We have:

`(x +3)^2 + 14(x + 3) + 49 = (U + V)^2`

Expand

`x^2 + 6x + 9 + 14x + 42 + 49 = (U + V)^2`

Collect like terms

`x^2 + 6x + 14x + 9 + 42 + 49 = (U + V)^2`

`x^2 + 20x + 100 = (U + V)^2`

Expand

`x^2 + 10x + 10x + 100 = (U + V)^2`

Group

`[x^2 + 10x] + [10x + 100] = (U + V)^2`

Factorize each group

`x[x + 10] + 10[x + 10] = (U + V)^2`

Factor out x + 10

`[x + 10][x + 10] = (U + V)^2`

So, we have:

`[x + 10]^2 = (U + V)^2`

By comparison

`U = x`

`V=10`