How does #t^2-2t-1=0# become #t=1+-sqrt2#?

Answer 1

#"see explanation"#

#"solve for t using the method of "color(blue)"completing the square"#
#• " the coefficient of the "t^2" term must be 1 which it is"#
#• " add "(1/2"coefficient of the t- term ")^2" to both sides"#
#"add 1 to both sides"#
#t^2-2t=1#
#t^2+2(-1)t color(red)(+1)=1color(red)(+1)#
#(t-1)^2=2#
#color(blue)"take the square root of both sides"#
#sqrt((t-1)^2)=+-sqrt2larrcolor(blue)"note plus or minus"#
#t-1=+-sqrt2#
#"add 1 to both sides to obtain"#
#t=1+-sqrt2#
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Answer 2

An alternate way using the quadratic formula:

Another way to find solutions for #t# is to use the quadratic formula:
# t = (-b \pm sqrt(b^2-4ac)) / (2a) #
with #a=1, b=-2, c=-1#
# t = (2 \pm sqrt((-2)^2-4(1)(-1))) / (2(1)) #
# t = (2 \pm sqrt(4+4)) / 2 #
# t = (2 \pm sqrt8) / 2 #
# t = (2 \pm 2sqrt2) / 2 #
# t = 1 \pm sqrt2 #
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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