Calculate the roots of -t^2-2t+1

[DottZakapa]

Homework Statement

-t^2-2t+1

Relevant Equations

roots are

I have to compute the roots in order to compute an integral partial fraction decomposition

$\frac {2 \pm 2 \sqrt {4+4}} {-2} = -1 \mp \sqrt 2$

the correct on is

$(t+1- \sqrt 2)(+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(+1- \sqrt 2)$

the general rule is

 

[PeroK]

Are you trying to factorise that quadratic expression?

 

[DottZakapa]

Are you trying to factorise that quadratic expression?

yes

 

[PeroK]

yes

The answer to the either/or is neither. There need to be two linear factors involving $t$. Such as $(t - r_1)(t-r_2)$ where $r_1, r_2$ are the roots.

If you think you have a factorisation, you can multiply it out to see whether you are correct.

 

[DottZakapa]

$\frac {2 \pm 2 \sqrt {2}} {-2} = -1 \mp \sqrt 2$
this is correct right?

 

[PeroK]

$\frac {2 \pm 2 \sqrt {4+4}} {-2} = -1 \mp \sqrt 2$
this is correct right?

No. It's arithmetically wrong, I'm sorry to say.

 

[DottZakapa]

sorry

No. It's arithmetically wrong, I'm sorry to say.

sorry i made a typing mistake, I've corrected the rest too
this is what i meant
$\frac {2 \pm 2 \sqrt {2}} {-2} = -1 \mp \sqrt 2$

 

[PeroK]

sorry

sorry i made a typing mistake, I've corrected the rest too
this is what i meant
$\frac {2 \pm 2 \sqrt {2}} {-2} = -1 \mp \sqrt 2$

These are now the roots of your quadratic.

 

[DottZakapa]

These are now the roots of your quadratic.

so between the two the correct is the second or the first
$(t+1- \sqrt 2)(+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(+1- \sqrt 2)$
that minus in front of r₁ multiplies both signs (the one in front of 1 and square root ) or just the one in front of one? that's my question

 

[PeroK]

so between the two the correct is the second or the first
$(t+1- \sqrt 2)(+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(+1- \sqrt 2)$
that minus in front of r₁ multiplies both signs (the one in front of 1 and square root ) or just the one in front of one? that's my question

If you put a $t$ in the second term, where it should be, then there is no difference.

$(t+1- \sqrt 2)(t+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(t+1- \sqrt 2)$

These are equal.

 

[DottZakapa]

If you put a $t$ in the second term, where it should be, then there is no difference.

$(t+1- \sqrt 2)(t+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(t+1- \sqrt 2)$

These are equal.

so if i multiply both or just the first doesn't make a difference

 

[PeroK]

so if i multiply both or just the first doesn't make a difference

Those two expressions are equal because it doesn't matter what order you multiply the factors. In general:

$(t-a)(t-b) = (t-b)(t-a)$

And, in general, for expressions involving numbers:

$XY = YX$

where $X$ and $Y$ can be any numbers or expressions involving numbers.

This property of numbers is called "commuting". It works for both addition and multiplication of numbers, as we also have:

$X + Y = Y + X$.

 

[kuruman]

If you put a $t$ in the second term, where it should be, then there is no difference.

$(t+1- \sqrt 2)(t+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(t+1- \sqrt 2)$

These are equal.

They may be equal, but neither is correct because they don't give the expected result when multiplied out. Note that the original expression is -t^2-2t+1. There is a negative sign in front of the quadratic term that is lost when the expresion is set equal to zero in order to find the roots. This doesn't affect the roots but it changes the overall sign in front of the desired partial fractions.

 

[DottZakapa]

They may be equal, but neither is correct because they don't give the expected result when multiplied out. Note that the original expression is -t^2-2t+1. There is a negative sign in front of the quadratic term that is lost when the expresion is set equal to zero in order to find the roots. This doesn't affect the roots but it changes the overall sign in front of the desired partial fractions.

could you please tell me exactly where the mistake is, I'm getting so confused with this equation

 

[PeroK]

could you please tell me exactly where the mistake is, I'm getting so confused with this equation

$-t^2 -2t +1 = - (t+1- \sqrt 2)(t+1+ \sqrt 2)$