Question

How to solve the given equation by “completing the square method”, \[{x^2} - 4x + 4 = 100\].

Answer 1

Hint: Any number that can be written in the form of ${a^2} + {b^2}$is “written in complete squares”. For example, we have an expression, ${x^2} + 4x + 20$, which is clearly not a square.We know that, ${(a + b)^2} = {a^2} + 2ab + {b^2}$ .So, the middle term can be broken up and written as, $4x = 2 \times 2 \times x$, where $a = x,b = 2$
By formula, ${b^2} = 4$ and the expression has a 20 as a constant. We can now write, ${x^2} + 4x + 20 = {x^2} + 2 \times 2 \times x + 4 - 4 + 20$.
In the above expression, we have added and subtracted 4 simultaneously.
$({x^2} + 4x + 4) + 16 = {(x + 2)^2} + {4^2}$
Thus, the expression is now in squared form.

Complete step by step answer:
\[{x^2} - 4x + 4 = 100\]
If we break the Left-hand side of the above equation, we get
\[{x^2} - (2 \times 2 \times x) + {2^2} = 100\]………… We can see that the LHS is already in the form of ${a^2} - 2ab + {b^2}$
$ \Rightarrow {(x - 2)^2} = {10^2}$……………… As we know ${(a - b)^2} = {a^2} - 2ab + {b^2}$
Since the exponents of the LHS and the RHS are same, we can write the equation as,
$ \Rightarrow (x - 2) = 10$
Thus, we now have a linear equation in hand, to be solved to get the final answer.
$ \therefore x = 10 + 2 = 12$

Hence, 12 is the final answer.

Note: A quadratic equation is one that is written in the form, $a{x^2} + bx + c = 0$. There are 3 ways one might use to solve a quadratic equation. Namely, mid-term Factoring, Using the quadratic formula and by completing the square. In any kind of question, you can use the formula, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$to find the factors.