solve for x: x2 +2x + 2 = 0
Answers
Answer:
X k power square h
Step-by-step explanation:
The question is x²−2x+2=0 . This quadratic equation has no real solutions as after completing the square we get:
(x)²+2.(x)(−1)+(−1)²+2−(−1)²=0
=>(x−1)²+2−(−1)²=0
=>(x−1)²+2−1=0
=>(x−1)²+1=0
=>(x−1)²=−1 [implying that the quadratic has complex solutions]
Another visual interpretation is that the equation corresponds to a parabola with it's line of symmetry at the line x=1 and is shifted upwards by 1 units. It's equation is:
y=(x−1)²+1
And we have to find out when the parabola touches the x-axis that is when the 'y-value' of the upward shifted parabola is 0 . To find that we replace y with 0 so that we can find out the 'x-value' which satisfies the equation.
Again, as the parabola is upward facing, that implies that it's lowest point is at the line of symmetry. Hence at x=1 , we have:
0=(1−1)²+1
=>0=0+1
=>0=1 [which is mathematically impossible]
Thus, we have found out that there are no real roots or solutions that satisfies the equation x²−2x+2=0
If you're okay with complex roots, then to find it out you can substitute ‘ −1 ' as ' i² '.Then the above equation would be:
x²−2x+2=0
=>(x−1)²+1=0
=>(x−1)²=−1
=>(x−1)²=i²
Raising both sides to the (1/2) power
=>(x−1)=±i
=>x=1±i [note the ± sign which implies that both the roots are complex conjugates of each other]