Find the roots of the equation x-3/x=2
Answers
Step-by-step explanation:
Let
f(x)=x3−x−2
To solve:
x3−x−2=0
As you have mentioned in the problem, you are aware of the nature of the roots. That is: 2 complex and 1 real. This can be found by studying the behaviour of the discriminant. The demand is to unearth these roots.
First let us determine if this real root is positive of negative. Notice
f(0)=−2 and f(2)=4
The function is continous and therefore must be cutting the x axis once. These values were chosen arbitrarily just to see if the REAL root is positive or negative. We have now determined: it is positive.
Subsititute
x=t+at
As you said you will use this for the future, this is what you must substitute into the depressed cubic with 1 real and 2 complex roots.
“a” and “t” are both constants. “t” is a constant chosen by us in order to make the expression simpler. You will see why later.
x3−x−2=0
(t+at)3−(t+at)−2=0
Expand this one:
t3+3ta+3a2t+a3t3−t−at−2=0
Let us collect like terms.
t3+t(3a−1)+1t(3a2−a)+a3t3−2=0
I want to chose such an “a” so that the middle two terms will cancel out in order for the expression to simplify.
(3a−1)=0∩(3a2−a)=0
a=1/3
The expression now becomes:
t3+a3t3−2=0
Multiply by t3 and substitute a=13
t6−2t3+(127)=0
Notice the above expression can be made as a quadratic by substituting t3=p
p2−2p+(127)=0
By the quadratic formula
p=9+(√78)9
Substitute all back to x
Recall:
p=9+(√78)9
t3=p
x=t+at
where
a=1/3
Answer:
x= -1
x= 3
Step-by-step explanation:
(x²-3)/x=2
x²-2x-3=0
x(x-3)+1(x-3)
x+1 , x-3
x= -1
x= 3
answer