Factorize completely X2+2XY+Y2+3X+3Y-18

Analysis

The expression to factorize is: $x^2 + 2xy + y^2 + 3x + 3y - 18$. This is a quadratic-like expression involving two variables ($x$ and $y$). We can simplify this by grouping terms and recognizing a perfect square trinomial.

Concept

1. Perfect Square Trinomial: Recall that $(x + y)^2 = x^2 + 2xy + y^2$.
2. Substitution: We can replace the group $(x + y)$ with a single variable (let's say $u$) to make the quadratic easier to factorize.
3. Factorization of Quadratics: Once in the form $u^2 + bu + c$, we look for two numbers that multiply to give $c$ and add to give $b$.

Step-by-Step Solution

Step 1: Group the terms. Notice that the first three terms form a perfect square: $(x^2 + 2xy + y^2) + 3x + 3y - 18$

Step 2: Simplify the expression. Substitute $(x + y)^2$ for the first part and factor out $3$ from the next two terms: $(x + y)^2 + 3(x + y) - 18$

Step 3: Let $u = (x + y)$. Now the expression becomes: $u^2 + 3u - 18$

Step 4: Factorize the quadratic. We need two numbers that multiply to $-18$ and add to $+3$. These numbers are $+6$ and $-3$. $(u + 6)(u - 3)$

Step 5: Substitute $(x + y)$ back in for $u$. Replace $u$ with $(x + y)$: $((x + y) + 6)((x + y) - 3)$ $= (x + y + 6)(x + y - 3)$

Conclusion

Comparing our result to the given options, we find that it matches option A.

Tip

Whenever you see an expression with $x^2, y^2,$ and $2xy$, always look for the $(x+y)^2$ or $(x-y)^2$ pattern first. It significantly reduces the complexity of the problem and saves you time during the JAMB exam!