Question

How do you solve $\sqrt {2x - 1} = x - 2$?

Answer 1

Hint: Take the given function and use the identity of square of two numbers and simplification of terms and making the required term “x” the subject and finding the value for it.

Complete step-by-step solution:
Take the given expression: $\sqrt {2x - 1} = x - 2$
Take square of the above expression on both the sides of the equation –
${\left( {\sqrt {2x - 1} } \right)^2} = {\left( {x - 2} \right)^2}$
Square and square-root cancel each other. Apply the identity for the difference of two terms and its whole square in the above expression.
$\Rightarrow 2x - 1 = {x^2} - 2x + 4$
The above expression can be re-written as –
$\Rightarrow {x^2} - 2x + 4 = 2x - 1$
Move all the terms on one side of the equation. When you move any term from one side of the equation to the opposite side then the sign of the terms also changes. Positive terms become negative and vice-versa.
$\Rightarrow {x^2} - 2x + 4 - 2x + 1 = 0$
Make the pair of like terms –
$\Rightarrow {x^2} - \underline {2x - 2x} + \underline {4 + 1} = 0$
Simplify the above expression –
$\Rightarrow {x^2} - 4x + 5 = 0$
Compare the above equation with the standard equation: $a{x^2} + bx + c = 0$
$
   \Rightarrow a = 1 \\
   \Rightarrow b = - 4 \\
   \Rightarrow c = 5 \\
$
Also, $\Delta = {b^2} - 4ac$
Place the values from the given comparison
$ \Rightarrow \Delta = {( - 4)^2} - 4(1)(5)$
Simplify the above equation –
$ \Rightarrow \Delta = 16 - 20$
$ \Rightarrow \Delta = - 4$
Take square root on both the sides of the equation –
\[ \Rightarrow \sqrt \Delta = \sqrt { - 4} = \sqrt {4( - 1)} \]
Simplify –
\[ \Rightarrow \sqrt \Delta = \pm 2i\]
Now, roots of the given equation can be expressed as –
\[\Rightarrow x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\]
Place values in the above equation –
\[\Rightarrow x = \dfrac{{ - ( - 4) \pm 2i}}{2}\]
 Simplify the above equation –
\[\Rightarrow x = \dfrac{{4 \pm 2i}}{2}\]
Common factors from the numerator and the denominator cancel each other.
Therefore \[x = 2 + 2i\]or \[x = 2 - 2i\]
This is the required solution.

Therefore the correct answer is \[x = 2 + 2i\]or \[x = 2 - 2i\]

Note: Be careful regarding the sign convention. Always remember that the square of negative number or the positive number is always positive. Also, product of two negative numbers is always positive whereas, product of one positive and one negative number gives us the negative number. Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $25{\text{ = 5 }} \times {\text{ 5, 25 = }}{{\text{5}}^2}$, generally it is denoted by n to the power two i.e. ${n^2}$.