Expand Using the Binomial Theorem (2x+5)^2

Problem

(2*x+5)2

Solution

1. Identify the terms and the exponent for the binomial expansion (a+b)n where a=2*x b=5 and n=2

2. Apply the Binomial Theorem formula, 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 each term from k=0 to k=2

(2*x+5)2=(2/0)(2*x)2*(5)0+(2/1)(2*x)1*(5)1+(2/2)(2*x)0*(5)2)))

4. Calculate the coefficients using the combination formula (n/k)=(n!)/(k!(n−k)!))

(2/0)=1)

(2/1)=2)

(2/2)=1)

5. Simplify each term by performing the exponentiation and multiplication.

(2*x+5)2=1*(4*x2)*(1)+2*(2*x)*(5)+1*(1)*(25)

(2*x+5)2=4*x2+20*x+25

Final Answer

(2*x+5)2=4*x2+20*x+25

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