Expand Using the Binomial Theorem (2x+3)^2

Problem

(2*x+3)2

Solution

1. Identify the terms and the exponent for the binomial expansion (a+b)n where a=2*x b=3 and n=2

2. Apply the formula for the Binomial Theorem, which states (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

3. Write the expansion by substituting the values into the formula for k=0,1,2

(2*x+3)2=(2/0)(2*x)2*(3)0+(2/1)(2*x)1*(3)1+(2/2)(2*x)0*(3)2)))

4. Evaluate the coefficients using the combination formula (n/k)=(n!)/(k!(n−k)!)) giving (2/0)=1) (2/1)=2) and (2/2)=1)

(2*x+3)2=1*(4*x2)*(1)+2*(2*x)*(3)+1*(1)*(9)

5. Simplify each term by performing the multiplication.

(2*x+3)2=4*x2+12*x+9

Final Answer

(2*x+3)2=4*x2+12*x+9

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