Simplify the following expression.
[tex]\begin{array}{c}
(3 x+7)^2 \\
{[?] x^2+\square x+\square}
\end{array}[/tex]
Answer (1)
Use the binomial expansion formula: ( a + b ) 2 = a 2 + 2 ab + b 2 .
Substitute a = 3 x and b = 7 into the formula: ( 3 x + 7 ) 2 = ( 3 x ) 2 + 2 ( 3 x ) ( 7 ) + ( 7 ) 2 .
Simplify each term: ( 3 x ) 2 = 9 x 2 , 2 ( 3 x ) ( 7 ) = 42 x , and ( 7 ) 2 = 49 .
The simplified expression is 9 x 2 + 42 x + 49 .
Explanation
Understanding the Problem We are given the expression ( 3 x + 7 ) 2 and asked to expand and simplify it. Our goal is to express it in the form a x 2 + b x + c , where a , b , and c are constants.
Applying the Binomial Expansion To expand the expression, we'll use the binomial expansion formula, which states that ( a + b ) 2 = a 2 + 2 ab + b 2 . In our case, a = 3 x and b = 7 .
Substituting the Values Now, let's substitute a = 3 x and b = 7 into the formula: ( 3 x + 7 ) 2 = ( 3 x ) 2 + 2 ( 3 x ) ( 7 ) + ( 7 ) 2
Simplifying the Expression Next, we simplify each term:
( 3 x ) 2 = 9 x 2
2 ( 3 x ) ( 7 ) = 42 x
( 7 ) 2 = 49
So, we have: ( 3 x + 7 ) 2 = 9 x 2 + 42 x + 49
Final Result Therefore, the simplified expression is 9 x 2 + 42 x + 49 . The coefficients are a = 9 , b = 42 , and c = 49 .
Examples
Understanding how to expand binomial expressions like ( 3 x + 7 ) 2 is useful in many real-world scenarios. For example, if you're planning a garden and want to calculate the area of a square plot with sides of length ( 3 x + 7 ) meters, you would need to expand this expression to find the area in terms of x . Similarly, in physics, you might encounter such expressions when dealing with projectile motion or wave functions. Knowing how to quickly and accurately expand these expressions can save time and prevent errors in your calculations.
