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Chapter 2: Problem 51

Expand the expression. $$ (3+\sqrt{x})^{2} $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: To expand the expression \((3+\sqrt{x})^{2}\), apply the binomial expansion formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\), with \(a=3\) and \(b=\sqrt{x}\). After calculating the individual terms and combining them, the expanded expression is \(9 + 6\sqrt{x} + x\).

Step by step solution

01

Recognize the Binomial

To expand the given expression, we need to recognize it as a binomial raised to a power. The binomial in this case is \((3+\sqrt{x})\) and it is raised to the power of 2.
02

Apply the Binomial Expansion Formula

In order to expand the expression, we will use the formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\), where \(a=3\) and \(b=\sqrt{x}\).
03

Calculate the Individual Terms

Using the formula from step 2, we will now calculate each term: - \(a^{2} = (3)^{2} = 9\) - \(2ab = 2(3)(\sqrt{x}) = 6\sqrt{x}\) - \(b^{2} = (\sqrt{x})^{2} = x\)
04

Combine the Terms

Now, we can combine the terms calculated in step 3 to form the expanded expression: \((3+\sqrt{x})^{2} = 9 + 6\sqrt{x} + x\) The expanded expression is \(9 + 6\sqrt{x} + x\).

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