Question

How do you complete the square to solve \[{x^2} - 4x - 5 = 0\] ?

Answer 1

Hint: We regroup the given equation. We have the equation in the form \[a{x^2} + bx + c = 0\] , we have the value of ‘b’ and dividing the value of ‘b’ by 2. We need to square the obtained value. Then we add and subtract the value in the given equation. We also use the formula \[{(a + b)^2} = {a^2} + 2ab + b\] . Using this we can solve the given problem.

Complete step-by-step answer:
Given, \[{x^2} - 4x - 5 = 0\] .
We regroup the equation we have,
 \[ \Rightarrow ({x^2} - 4x) - 5 = 0\] .
The given equation is in the form \[a{x^2} + bx + c = 0\] , where \[a = 1\] and \[b = - 4\] .
We divide ‘b’ by two,
 \[ \Rightarrow \dfrac{{ - 4}}{2} = - 2\]
Now we need to square the obtained value that is \[ \Rightarrow {( - 2)^2} = 4\] .
We add this answer inside the parenthesis
 \[ \Rightarrow ({x^2} - 4x + 4) - 5 = 0\]
 Since we add four to the original equation, we need to adjust it by subtracting it by four.
 \[ \Rightarrow ({x^2} - 4x + 4) - 5 - 4 = 0\]
 \[ \Rightarrow ({x^2} - 4x + 4) - 9 = 0\]
Now we can see that \[{x^2} - 4x + 4\] is in the form \[{a^2} + 2ab + {b^2}\] with \[a = x\] and \[b = - 2\] .
That is we can rewrite this into the form \[{(a + b)^2}\] . That is \[{(x - 2)^2}\] .
 \[ \Rightarrow {(x - 2)^2} - 9 = 0\]
Adding 9 on both sides we get,
 \[ \Rightarrow {(x - 2)^2} = 9\]
Taking square root on both side and we know square and square root cancels out,
 \[ \Rightarrow (x - 2) = \sqrt 9 \]
 \[ \Rightarrow x - 2 = \pm 3\]
Thus we have two roots,
 \[ \Rightarrow x - 2 = 3\] and \[x - 2 = - 3\] .
 \[ \Rightarrow x = 3 + 2\] and \[x = - 3 + 2\]
 \[ \Rightarrow x = 5\] and \[x = - 1\] .
So, the correct answer is “ \[ x = 5\] and \[x = - 1\] ”.

Note: We follow the same procedure for these kinds of problems. Careful in the calculation part. Careful while adding the value after squaring (see above), we need to balance the equation by subtracting the same number in the equation. We also solve this by factorization. But this is not what they asked in the question. If we do by the factorization we get the same answer but It is wrong.