Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's simplify and expand the expression [tex]\(\left(2u + \frac{3}{u} - 1\right)^2\)[/tex] step-by-step.
### Step 1: Understand the Expression Inside the Square
We have the expression inside the square as [tex]\(2u + \frac{3}{u} - 1\)[/tex].
### Step 2: Apply the Square Formula
Recall the algebraic identity for squaring a trinomial:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \][/tex]
In our case, [tex]\(a = 2u\)[/tex], [tex]\(b = \frac{3}{u}\)[/tex], and [tex]\(c = -1\)[/tex]. So we can write:
[tex]\[ (2u + \frac{3}{u} - 1)^2 \][/tex]
### Step 3: Compute Individual Squares and Cross-terms
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[ (2u)^2 = 4u^2 \][/tex]
2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[ \left(\frac{3}{u}\right)^2 = \frac{9}{u^2} \][/tex]
3. Calculate [tex]\(c^2\)[/tex]:
[tex]\[ (-1)^2 = 1 \][/tex]
4. Calculate [tex]\(2ab\)[/tex]:
[tex]\[ 2(2u) \left(\frac{3}{u}\right) = 2 \cdot 2u \cdot \frac{3}{u} = 2 \cdot 6 = 6 \][/tex]
5. Calculate [tex]\(2ac\)[/tex]:
[tex]\[ 2(2u) \cdot (-1) = 2 \cdot 2u \cdot -1 = -4u \][/tex]
6. Calculate [tex]\(2bc\)[/tex]:
[tex]\[ 2 \left( \frac{3}{u} \right) \cdot (-1) = 2 \cdot \frac{3}{u} \cdot -1 = -\frac{6}{u} \][/tex]
### Step 4: Sum up All the Parts
Combining all these results together, we get the expanded expression:
[tex]\[ 4u^2 + \frac{9}{u^2} + 1 + 6 - 4u - \frac{6}{u} \][/tex]
### Step 5: Simplify the Expression
Collect and combine like terms:
[tex]\[ 4u^2 - 4u + 1 + 6 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
Thus, the expanded form of [tex]\(\left(2u + \frac{3}{u} - 1\right)^2\)[/tex] is:
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
### Conclusion
We can summarize our results as the original and expanded forms:
[tex]\[ \left(2u + \frac{3}{u} - 1\right)^2 = \left(2u + \frac{3}{u} - 1\right)^2 \][/tex]
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
So, the original expression [tex]\(\left(2u + \frac{3}{u} - 1\right)^2\)[/tex] simplifies and expands to:
[tex]\[ (2u - 1 + \frac{3}{u})^2 = 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
### Step 1: Understand the Expression Inside the Square
We have the expression inside the square as [tex]\(2u + \frac{3}{u} - 1\)[/tex].
### Step 2: Apply the Square Formula
Recall the algebraic identity for squaring a trinomial:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \][/tex]
In our case, [tex]\(a = 2u\)[/tex], [tex]\(b = \frac{3}{u}\)[/tex], and [tex]\(c = -1\)[/tex]. So we can write:
[tex]\[ (2u + \frac{3}{u} - 1)^2 \][/tex]
### Step 3: Compute Individual Squares and Cross-terms
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[ (2u)^2 = 4u^2 \][/tex]
2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[ \left(\frac{3}{u}\right)^2 = \frac{9}{u^2} \][/tex]
3. Calculate [tex]\(c^2\)[/tex]:
[tex]\[ (-1)^2 = 1 \][/tex]
4. Calculate [tex]\(2ab\)[/tex]:
[tex]\[ 2(2u) \left(\frac{3}{u}\right) = 2 \cdot 2u \cdot \frac{3}{u} = 2 \cdot 6 = 6 \][/tex]
5. Calculate [tex]\(2ac\)[/tex]:
[tex]\[ 2(2u) \cdot (-1) = 2 \cdot 2u \cdot -1 = -4u \][/tex]
6. Calculate [tex]\(2bc\)[/tex]:
[tex]\[ 2 \left( \frac{3}{u} \right) \cdot (-1) = 2 \cdot \frac{3}{u} \cdot -1 = -\frac{6}{u} \][/tex]
### Step 4: Sum up All the Parts
Combining all these results together, we get the expanded expression:
[tex]\[ 4u^2 + \frac{9}{u^2} + 1 + 6 - 4u - \frac{6}{u} \][/tex]
### Step 5: Simplify the Expression
Collect and combine like terms:
[tex]\[ 4u^2 - 4u + 1 + 6 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
Thus, the expanded form of [tex]\(\left(2u + \frac{3}{u} - 1\right)^2\)[/tex] is:
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
### Conclusion
We can summarize our results as the original and expanded forms:
[tex]\[ \left(2u + \frac{3}{u} - 1\right)^2 = \left(2u + \frac{3}{u} - 1\right)^2 \][/tex]
[tex]\[ 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
So, the original expression [tex]\(\left(2u + \frac{3}{u} - 1\right)^2\)[/tex] simplifies and expands to:
[tex]\[ (2u - 1 + \frac{3}{u})^2 = 4u^2 - 4u + 7 - \frac{6}{u} + \frac{9}{u^2} \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
