Simplify this equation: -(t - 3)^2 + 6(t - 3) - 11

[Saturnize]

Help Me simplify this equation... I need help with this

. . . . .-(t - 3)² + 6(t - 3) - 11

I already know what the answer is, but no matter what website I go to; there always seems to be
a lack of explaination as to how to simplify it.

For example

https://www.mathpapa.com/algebra-calculator.html

gives a step by step explaination, but random numbers seem to pop out of nowhere.
I used FOIL manage the squared paranthesis.

Please help Me and give a spoonfed, step by step explaination on how You solved this. ^^

 

[Dr.Peterson]

I need help with this

View attachment 10310

I already know what the answer is, but no matter what website I go to; there always seems to be
a lack of explanation as to how to simplify it.

For example

https://www.mathpapa.com/algebra-calculator.html

gives a step by step explaination, but random numbers seem to pop out of nowhere.
I used FOIL manage the squared paranthesis.

Please help Me and give a spoonfed, step by step explaination on how You solved this. ^^

The expression (it isn't an equation!), written as we do in type, is

-(t - 3)^2 + 6(t - 3) - 11
​

Here is what that site tells you:

Let's simplify step-by-step.
​

−(t−3)²+6(t−3)−11
​

Distribute:​

=−t²+6t+−9+(6)(t)+(6)(−3)+−11
​

=−t²+6t+−9+6t+−18+−11
​

Combine Like Terms:​

=−t²+6t+−9+6t+−18+−11
​

=(−t²)+(6t+6t)+(−9+−18+−11)
​

=−t²+12t+−38
​

---

Answer:​

=−t²+12t−38
​

They are assuming you can handle the individual steps, because if they always showed every possible detail, they would overwhelm their readers. (Spoon-feeding, taken to the extreme, leads to choking!) This is why math is taught in stages, hopefully getting you to master one process (e.g. combining like terms, or adding signed numbers) before doing a bigger process that uses it. They don't know where you are in the learning process, so they can't aim their work directly at your needs. Nor can I, without input!

The one I think is most likely to confuse you is the first, where they expand -(t-3)² as -t² + 6t + -9. Here it is, taken very slowly:

-(t-3)² = -(t-3)(t-3) = -(t² - 3t - 3t + 9) [applying FOIL to (t-3)(t-3)]
= -(t² - 6t + 9) [combining like terms]
= -t² -(-6t) - 9 [distributing the negation, which multiplies each term by -1]
= -t² + 6t - 9 [negative of negative is positive]
​

But you seem to be saying that you are okay with that step.

In order to be sure to help you, I need to know where your specific issues are. If I tried to show every single detail, I would only make it look more confusing, and probably miss your own issue, just as that site has done.

Please point to particular places where you don't know what is being done or where a number comes from. Ask specific questions, and we should be able to help. That's the benefit of human interaction over computers. But to do this well, we'll have to actually interact.

 

[Denis]

-(t - 3)^2 + 6(t - 3) - 11

If the 1st term is kinda confusing you, pretend the leading - sign ain't there:
(t - 3)^2
= (t - 3)*(t - 3)
= t^2 - 6t + 9
Now multiply by -1:
-t^2 + 6t - 9

 

[JeffM]

I have not looked at the site you referenced and am relying on Dr. P's research, but

−(t−3)²+6(t−3)−11
​

Distribute:​

=−t²+6t+−9+(6)(t)+(6)(−3)+−11
​

=−t²+6t+−9+6t+−18+−11
​

Combine Like Terms:​

=−t²+6t+−9+6t+−18+−11
​

=(−t²)+(6t+6t)+(−9+−18+−11)
​

=−t²+12t+−38
​

---

Answer:​

=−t²+12t−38​

That does not strike me as the clearest possible explanation. I hate that +- stuff (it is not wrong, merely distracting). Moreover, I like to deal with primarily positive expressions when possible so I would go:

\(\displaystyle -\ (t - 3)^2 + 6(t - 3) - 11 =\)

\(\displaystyle (-\ 1)\{(t - 3)^2 - 6(t - 3) + 11 \} =\)

\(\displaystyle (-\ 1)\{t^2 - 6t + 9 - 6(t - 3) + 11\} =\)

\(\displaystyle (-\ 1)\{t^2 - 6t + 20 - 6(t - 3)\} =\)

\(\displaystyle (-\ 1)\{t^2 - 6t + 20 - 6t - 6(-\ 3)\} =\)

\(\displaystyle (-\ 1)(t^2 - 6t + 20 - 6t + 18) =\)

\(\displaystyle (-\ 1)(t^2 - 12t + 38) =\)

\(\displaystyle -\ t^2 + 12t - 38.\)

So in my 2nd line, I get an expression that has mostly positive terms. (This step is not necessary; I just find that I make fewer silly mistakes if I don't have minus signs all over the place.)

In the third line, I simply expand the squared term.

In the fourth line, I do some preliminary simplification.

In the fifth line, I distribute the minus 6 over t and minus 3.

In the sixth line, I finish the arithmetic multiplication.

In the seventh line, I combine like terms.

In the eighth line, I distribute the minus 1 over the quadratic polynomial.

Now of course I probably would not do this step by step in practice; it is slow.

But what I would do is to check my work with a small number not equal to 0 or 1.

If t = 5

\(\displaystyle - (5 - 3)^2 + 6(5 - 3) - 11 = -\ 2^2 + 6 * 2 - 11 = -\ 4 + 12 - 11 = -\ 3.\)

\(\displaystyle -\ 5^2 + 12 * 5 - 38 = -\ 25 + 60 - 38 = 60 - 63 = -\ 3.\)

So it checks. But if it did not check, then I probably would do baby steps to find my error.

 

[lookagain]

But what I would do is to check my work with a small number not equal to 0 or 1.

If t = 5

\(\displaystyle - (5 - 3)^2 + 6(5 - 3) - 11 = -\ 2^2 + 6 * 2 - 11 = -\ 4 + 12 - 11 = -\ 3.\)

\(\displaystyle -\ 5^2 + 12 * 5 - 38 = -\ 25 + 60 - 38 = 60 - 63 = -\ 3.\)

So it checks. But if it did not check, then I probably would do baby steps to find my error.

The fact that t = 5 checks is promising, but it does not mean that trinomial is correct.
If a second t-value were to check, that would also not guarantee it. It would take a
minimum of three t-values to check to show that that resultant quadratic polynomial
is equivalent to the original expression.

____________________________________________

\(\displaystyle -(t - 3)^2 + 6(t - 3) - 11 = \)

\(\displaystyle -[(t - 3)^2 - 6(t - 3) \ + \ 11] = \)

\(\displaystyle -[(t - 3)(t - 3 \ \ - \ 6) \ + \ 11] =\)

\(\displaystyle -[(t - 3)(t - 9) \ + \ 11] = \)

\(\displaystyle -[t^2 - 9t - 3t + 27 + 11] =\)

\(\displaystyle -(t^2 - 12t + 38) =\)

\(\displaystyle -t^2 + 12t - 38\)

 

[JeffM]

The fact that t = 5 checks is promising, but it does not mean that trinomial is correct.
If a second t-value were to check, that would also not guarantee it. It would take a
minimum of three t-values to check to show that that resultant quadratic polynomial
is equivalent to the original expression.

You are correct.

If we had to prove that the two expressions were equivalent, a single point would not suffice. But the probability that a number essentially chosen out of the blue will check when the expressions are not equivalent is tiny. Getting students to do any checking is hard; I doubt it is productive to insist on true proof. Nor is it worthwhile on a timed test. In fact, I have found that very many students have not been taught that it is even possible to check numerically whether such a derivation has a high probability of being correct.

But it would indeed be more precise to indicate that such a check merely provides a high probability that the two expressions are indeed equivalent.

 

[Dr.Peterson]

You are correct.

If we had to prove that the two expressions were equivalent, a single point would not suffice. But the probability that a number essentially chosen out of the blue will check when the expressions are not equivalent is tiny. Getting students to do any checking is hard; I doubt it is productive to insist on true proof. Nor is it worthwhile on a timed test. In fact, I have found that very many students have not been taught that it is even possible to check numerically whether such a derivation has a high probability of being correct.

But it would indeed be more precise to indicate that such a check merely provides a high probability that the two expressions are indeed equivalent.

I agree fully.

I describe such a check as a "spot check" ("a test made without warning on a randomly selected subject") -- not a proof of correctness, but an encouragement that your answer is reasonable, or a warning that it is not.

 

[Saturnize]

Ok, thanks for the feedback everyone. I wasn't aware that You were to distribute the negative sign in such a fashion. Even so I should have simply got an inverted answer and not

t^2+6t-38

Which isn't even an inverted answer... I think I just did a step too early,s kipped one, misprinted something, ect. -_-

I'm going to do more examples to make sure it sticks.