Question

Solve by completing the square method \[x_{}^2 - 2x - 30 = 0\]

Answer 1

Hint: In order to solve the question we need to add $1$ on both the sides of the given equation at the very first stage.
Then we use the formula for squaring the terms.
Finally we get the required answer.

Formula used: $(a - b)_{}^2 = a_{}^2 - 2ab + b_{}^2$

Complete step-by-step answer:
It is given in the question that \[x_{}^2 - 2x - 30 = 0\]
To solve the equation we have to move $30$ on the right hand side and we get-
$\Rightarrow$$x_{}^2 - 2x = 30$
Here we have to apply square method, we need to add $1$ on both the sides of the equation and we get-
$\Rightarrow$$x_{}^2 - 2x + 1 = 30 + 1$
In the Left Hand Side, we compared that the general quadratic equation $ax_{}^2 - bx + c$
We can write it as, $x_{}^2 - 2x + 1$
Here\[a = 1\],\[\;b = - 2\] and \[c = 1\]
Now we can apply the formula of $(a - b)_{}^2 = a_{}^2 - 2ab + b_{}^2$ and we get
$\Rightarrow$$(x - 1)_{}^2 = 31$
By applying square root on both sides we get,
$\Rightarrow$$x - 1 = \sqrt {31} $
To find out the value of $x$ we need to move $1$on the right hand side and we get-
$\Rightarrow$$x = 1 + \sqrt {31} $
As we know that a square root has two solutions.
Therefore, the equation \[x_{}^2 - 2x - 30 = 0\] will have two values, one is positive and the other one is negative.
Therefore the required values of \[x_{}^2 - 2x - 30 = 0\] are $1 + \sqrt {31} $ and $1 - \sqrt {31} $

Hence by completion of the square method in the given equations, roots are $1 + \sqrt {31} $ and $1 - \sqrt {31} $

Note: Since it is mentioned in the question that we have to solve it by a squaring method so we have applied it otherwise this quadratic equation can also be solved by applying the shree Acharya formula.
While solving the question through the squaring method, keep in mind that square root gives one positive and one negative value because many of the students make mistakes here.