Simplifying f(a+h)=-5(a+h)^2+2(a+h)-1

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In summary, the "freshman's dream" is when you think that:(x+y)^n=x^n+b^nis a mistake so commonly made by students, it's been given a name...the "freshman's dream." :)
  • #1
Rujaxso
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Not sure where the 2ah is coming from in the middle step.

f(a+h)=-5(a+h)^2+2(a+h)-1 = -5(a^2 + 2ah +h^2) +2a +2h -1 = -5a^2 - 10ah -5h^2 +2a +2h -1

Please enlighten me
 
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  • #2
Rujaxso said:
Not sure where the 2ah is coming from in the middle step.

f(a+h)=-5(a+h)^2+2(a+h)-1 = -5(a^2 + 2ah +h^2) +2a +2h -1 = -5a^2 - 10ah -5h^2 +2a +2h -1

Please enlighten me

When you square a binomial, the following rule applies:

\(\displaystyle (x+y)^2=x^2+2xy+y^2\)

Have you seen this rule before?
 
  • #3
Thanks Mark.
Nope I haven't, I was just going off of what I know about distributing.
 
  • #4
And here is how you would do that distribution: $(a+ h)^2= (a+ h)(a+ h)= a(a+ h)+ h(a+ h)= a^2+ ah+ ha+ h^2= a^2+ 2ah+ h^2$.
 
  • #5
Okay so instead of squaring each term I need to think about the quantity as a whole in the parenthesis as being a base for the exponent?
 
  • #6
MarkFL said:
When you square a binomial, the following rule applies:

\(\displaystyle (x+y)^2=x^2+2xy+y^2\)

Have you seen this rule before?

Btw
Found this under Algebra 1 > polynomials > special products on khan academy, ...I will go over that section
 
  • #7
Rujaxso said:
Okay so instead of squaring each term I need to think about the quantity as a whole in the parenthesis as being a base for the exponent?
Yes, that's what the parentheses mean. $( )^2$ means you square whatever is in the parentheses. $( )^3$ means you cube what ever is in the parentheses. $\sin( )$ means you take the sine of whatever is in the parentheses. In general $f( )$ means you apply the function f to whatever is in the parentheses.
 
  • #8
I guess I wanted to apply A(B+C) = AB + AC to (B+C)^2 and make B^2 + C^2 which is incorrect I see.
 
  • #9
Rujaxso said:
I guess I wanted to apply A(B+C) = AB + AC to (B+C)^2 and make B^2 + C^2 which is incorrect I see.

Thinking that:

\(\displaystyle (x+y)^n=x^n+b^n\)

is a mistake so commonly made by students, it's been given a name...the "freshman's dream." :)

I have also seen a lot of students make a related mistake, and that is to state:

\(\displaystyle \sqrt{x^2+y^2}=x+y\)
 

What is the purpose of simplifying f(a+h)=-5(a+h)^2+2(a+h)-1?

The purpose of simplifying this equation is to make it easier to understand and work with. By simplifying, we can identify patterns and relationships and use them to solve problems or find solutions.

What are the steps involved in simplifying f(a+h)=-5(a+h)^2+2(a+h)-1?

The steps involved in simplifying this equation include expanding the brackets, combining like terms, and simplifying any remaining terms. This will result in a simpler form of the equation that is easier to work with.

Why is it important to simplify equations like f(a+h)=-5(a+h)^2+2(a+h)-1?

Simplifying equations is important because it allows us to solve problems and make predictions based on the underlying patterns and relationships. It also helps us to identify the key variables and understand how changes in those variables affect the overall equation.

What are some common mistakes to avoid when simplifying f(a+h)=-5(a+h)^2+2(a+h)-1?

Some common mistakes to avoid when simplifying this equation include not correctly expanding the brackets, forgetting to combine like terms, or making errors when simplifying the remaining terms. It is important to double check every step and make sure that the equation is being simplified correctly.

How can simplifying f(a+h)=-5(a+h)^2+2(a+h)-1 be useful in real-world applications?

Simplifying equations like this one can be useful in real-world applications, such as in physics, engineering, and economics. By understanding the underlying patterns and relationships, we can use these equations to make predictions and solve problems in various fields.

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