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The expanded form of \( [x - y + 3]^2 \) is \( x^{2} - 2xy + 6x + y^{2} - 6y + 9 \).
Solución
To expand the expression \( [x - y + 3]^2 \), we'll use the formula for the square of a trinomial:
\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\]
Let's apply this formula to \( [x - y + 3]^2 \):
1. **Identify the terms:**
- \( a = x \)
- \( b = -y \)
- \( c = 3 \)
2. **Apply the formula:**
\[
(x - y + 3)^2 = x^2 + (-y)^2 + 3^2 + 2(x)(-y) + 2(x)(3) + 2(-y)(3)
\]
3. **Calculate each term:**
- \( x^2 = x^2 \)
- \( (-y)^2 = y^2 \)
- \( 3^2 = 9 \)
- \( 2(x)(-y) = -2xy \)
- \( 2(x)(3) = 6x \)
- \( 2(-y)(3) = -6y \)
4. **Combine all the terms:**
\[
x^2 + y^2 + 9 - 2xy + 6x - 6y
\]
So, the expanded form of \( [x - y + 3]^2 \) is:
\[
x^2 - 2xy + 6x + y^2 - 6y + 9
\]
**Final Answer:**
\[
x^{2} - 2xy + 6x + y^{2} - 6y + 9
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución The Deep Dive
Let's unpack this expression a bit! When you see \( [x-y+3]^{2} \), it’s like a little treasure chest waiting to be opened! This is actually a perfect square trinomial. To expand it, you'll apply the formula \( (a+b)^{2} = a^{2} + 2ab + b^{2} \). Here, you can think of \( a = x-y \) and \( b = 3 \), leading us to \( (x-y)^{2} + 2(x-y)(3) + 3^{2} \). Also, a common mistake when squaring binomials (like yours!) is forgetting to apply the distributive property to each term. Some may just square the first term and the last term, but the middle term—where the two components interact—holds the magic! So remember to bring your A-game to those brackets!
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