Question

Solve the following equation:
$ - {x^2} + x - 2 = 0$
$\left( a \right){\text{ }}\dfrac{{1 \pm \sqrt 7 i}}{2}$
$\left( b \right){\text{ }}\dfrac{{ - 1 \pm \sqrt 7 i}}{2}$
$\left( c \right){\text{ }}\dfrac{{2 \pm \sqrt 7 i}}{2}$
$\left( d \right){\text{ }}\dfrac{{ - 2 \pm \sqrt 7 i}}{2}$

Answer 1

Hint:
This question will be solved by using the quadratic formula which is $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
From this we will have the values of the constants from the equation and putting in the formula we will get the solution for it.

Formula used:
Quadratic formula
Suppose an equation given as $a{x^2} + bx + c$ then the value for the $x$ will be given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here, $a, b, c$ are the variables and the $x$ will be the constant term.

Complete step by step solution:
First of all we will check or identify the equation whether it is the standard form or not. Since the equation is in standard form so we will note down the values$a,b,c$.$ - {x^2} + x - 2 = 0$
Therefore, $a = - 1,b = 1,c = - 2$
By utilizing the equation and subbing the values, the new equation will be as
$ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {{1^2} - 4\left( { - 1} \right)\left( { - 2} \right)} }}{{2\left( { - 1} \right)}}$
Now on evaluating the exponent, we get the equation as
$ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {1 - 4\left( { - 1} \right)\left( { - 2} \right)} }}{{2\left( { - 1} \right)}}$
So now on multiplying the numbers, we get
$ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {1 - 8} }}{{ - 2}}$
On subtracting the above numbers, we get
$ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt { - 7} }}{{ - 2}}$
As we know that the square root of the number which is real is not a real number. Therefore, the discriminant for this equation is negative so there will be no real equations, so the above equation can also be written as
On taking the negative sign common, we get
$ \Rightarrow x = \dfrac{{1 \pm \sqrt 7 i}}{2}$

Therefore, the option $\left( a \right)$ is correct.

Note:
This type of question can be solved in more than one way. We can also solve it by using the factorization method. One more way to solve the equation is by using the graphical method we can solve such equations. By completing the square method also we can solve it. Therefore, while solving it we just have to be aware of the equation as minor mistakes will harm the whole solution.