Simplify: $-3(y+2)^2-5+6 y$
What is the simplified product in standard form?
$\square y^2 + \square y + \square$
Answer (2)
Expand the squared term: ( y + 2 ) 2 = y 2 + 4 y + 4 .
Distribute the -3: − 3 ( y 2 + 4 y + 4 ) = − 3 y 2 − 12 y − 12 .
Combine like terms: − 3 y 2 − 12 y − 12 − 5 + 6 y = − 3 y 2 − 6 y − 17 .
The simplified expression is − 3 y 2 − 6 y − 17 .
Explanation
Understanding the Problem We are given the expression − 3 ( y + 2 ) 2 − 5 + 6 y and asked to simplify it and write it in the standard form a y 2 + b y + c . This means we need to expand the squared term, distribute the constants, and combine like terms to get the expression into the desired form.
Expanding the Squared Term First, let's expand the squared term ( y + 2 ) 2 . We have: ( y + 2 ) 2 = ( y + 2 ) ( y + 2 ) = y 2 + 2 y + 2 y + 4 = y 2 + 4 y + 4
Substituting Back Now, we substitute this back into the original expression: − 3 ( y + 2 ) 2 − 5 + 6 y = − 3 ( y 2 + 4 y + 4 ) − 5 + 6 y
Distributing the Constant Next, distribute the -3 across the terms inside the parentheses: − 3 ( y 2 + 4 y + 4 ) = − 3 y 2 − 12 y − 12
Substituting Again Now, substitute this back into the expression: − 3 y 2 − 12 y − 12 − 5 + 6 y
Combining Like Terms Finally, combine like terms: − 3 y 2 + ( − 12 y + 6 y ) + ( − 12 − 5 ) = − 3 y 2 − 6 y − 17
Final Answer So, the simplified expression in standard form is − 3 y 2 − 6 y − 17 . Therefore, the coefficients are: a = − 3 , b = − 6 , and c = − 17 .
Thus, the simplified product in standard form is: − 3 y 2 + − 6 y + − 17
Examples
Understanding how to simplify algebraic expressions like this is crucial in many areas, such as physics and engineering. For instance, when calculating the trajectory of a projectile, you often need to simplify complex equations to predict its path. Similarly, in electrical engineering, simplifying circuit equations helps in designing efficient and reliable systems. Mastering these simplification techniques provides a strong foundation for solving real-world problems in various scientific and technical fields.
To simplify − 3 ( y + 2 ) 2 − 5 + 6 y , we expand the squared term, distribute, and combine like terms, which results in − 3 y 2 − 6 y − 17 . The answer is in standard form with coefficients − 3 , − 6 , and − 17 . Therefore, the final form is − 3 y 2 − 6 y − 17 .
;
