Why are these two statements equivalent to each other? k(k+1) + 2(k+1)\\= (k+2) * (k+1)\\

Question

Why are these two statements equivalent to each other?


`k(k+1) + 2(k+1)\\= (k+2) * (k+1)\\`

Answer 1

Hello!

Answer:

The two statements are equivalent because they can be simplified to the same expression.

Explanation:

▪ We want to find why are these two statements equivalent to each other.

▪ We know that
`\sf a(b+c) + d(b+c)` is equal to
`\sf (a+d)(b+c)` since
`\sf (b + c)` is the factor common to both pairs.

▪ In our expression, the factor common to both pairs is
`\sf (k + 1)`.

▪ So the expression is equal to:

`\sf (k+2)(k+1)`

▪ So the two expressions can be simplified to the same expression.

▪ So they are equivalent.

Conclusion:

The two statements are equivalent because they can be simplified to the same expression.

So, the expression k(k + 1) + 2(k + 1) is equal to (k + 2)(k + 1).