Question

Solve by completing the square method,
\[{x^2} - 2x - 80 = 0\]

Answer 1

Hint: Here since it is given that we have to solve the given equation completing the square method so we will try to make the perfect square by adding and subtracting certain values and evaluating the value of x.

Complete step-by-step answer:
The given equation is:-
 \[{x^2} - 2x - 80 = 0\]
Now adding and subtracting 1 we get:-
\[{x^2} - 2x - 80 + 1 - 1 = 0\]
Rearranging it we get:-
\[\left( {{x^2} - 2x + 1} \right) - 80 - 1 = 0\]
Now we know that:-
\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
Applying this identity we get:-
\[{\left( {x - 1} \right)^2} - 80 - 1 = 0\]
Solving it further we get:-
\[{\left( {x - 1} \right)^2} - 81 = 0\]
\[ \Rightarrow {\left( {x - 1} \right)^2} = 81\]
Now taking square root both the sides we get:-
\[\sqrt {{{\left( {x - 1} \right)}^2}} = \sqrt {81} \]
Simplifying it we get:-
\[x - 1 = \pm 9\]
Hence, \[x - 1 = 9;x - 1 = - 9\]
\[x = 10;x = - 8\]
Therefore,.
Therefore values of x are 10 and -8.

Note: Students can verify their answer by solving the given problem by splitting the middle term method.
The given equation is:-
\[{x^2} - 2x - 80 = 0\]
Applying middle term split we get:-
\[{x^2} - 10x + 8x - 80 = 0\]
Taking terms common we get:-
\[x\left( {x - 10} \right) + 8\left( {x - 10} \right) = 0\]
Simplifying it further we get:-
\[\left( {x + 8} \right)\left( {x - 10} \right) = 0\]
Solving for x we get:-
\[x = - 8;x = 10\]