Question

How do you solve \[ - {x^2} + 2x - 1 = 0\] ?

Answer 1

Hint: Given is a quadratic equation. First we will multiply both the sides by minus sign to remove the minus sign of the first term of the given equation. Then we can factorize the middle term or can use quadratic formula but since after multiplication we can get that the quadratic equation becomes a perfect square. So we will just take the root of both sides and then find the answer.

Complete step-by-step answer:
Given that,
 \[ - {x^2} + 2x - 1 = 0\]
Multiply both sides by minus sign we get,
 \[{x^2} - 2x + 1 = 0\]
The signs are just reversed. Now we can observe that the equation now is of the form \[{a^2} - 2ab + {b^2} = 0\]
So this is nothing but the identity for \[{a^2} - 2ab + {b^2} = {\left( {a - b} \right)^2}\]
So we can write the given expression as,
 \[{\left( {x - 1} \right)^2} = 0\]
Now taking square root on both sides,
 \[x - 1 = 0\]
Taking 1 on other side of equation,
 \[x = 1\]
This is our answer.
So, the correct answer is “x = 1”.

Note: Here note that the equation should have two roots. So the two roots here are same and equal. To cross verify this let’s find the discriminant.
Discriminant is value of \[\sqrt {{b^2} - 4ac} \]
 \[\sqrt {{b^2} - 4ac} = \sqrt {{2^2} - 4 \times 1 \times 1} = \sqrt {4 - 4} = 0\]
So we know that if the discriminant is zero then the roots are the same and equal.
This is important if we are answering multiple choice questions. One should not get confused with the other root. Is it plus or minus? For that we can take help of discriminant.