Question

How do you solve $ \sqrt {x + 2} - x = 0 $

Answer 1

Hint: In this question, we are given an algebraic expression as the given expression is the combination of alphabet “x” and the numerical values. They are linked by some arithmetic operations like addition, subtraction, multiplication and division. We know that for finding the values of “n” unknown variables, we need an “n” number of algebraic expressions. Here we have exactly 1 algebraic equation and 1 unknown variable “x”. So, we can solve it easily by first taking “x” to the right-hand side and then squaring both sides. Now, we will get a quadratic equation that will be solved by using factorization or the quadratic formula.

Complete step by step solution:
We are given $ \sqrt {x + 2} - x = 0 $
Take “x” to the right-hand side –
 $ \sqrt {x + 2} = x $
Squaring both the sides, we get –
\[
  {(\sqrt {x + 2} )^2} = {x^2} \\
   \Rightarrow {x^2} = x + 2 \\
   \Rightarrow {x^2} - x - 2 = 0 \;
 \]
Now, the above equation has 2 as the highest exponent of “x” so it is an equation of degree 2, and is thus known as the quadratic equation. It can be solved by using factorization as follows –
 $
  {x^2} - 2x + x - 2 = 0 \\
   \Rightarrow x(x - 2) + 1(x - 2) = 0 \\
   \Rightarrow (x + 1)(x - 2) = 0 \\
   \Rightarrow x + 1 = 0,\,x - 2 = 0 \\
   \Rightarrow x = - 1,\,x = 2 \;
  $
Hence when $ \sqrt {x + 2} - x = 0 $ , we get $ x = - 1,\,x - 2 $ .
So, the correct answer is “ $ x = - 1,\,x - 2 $ ”.

Note: When the unknown variable quantity in an algebraic expression is raised to some non-negative integer as its power, then it is known as the polynomial equation. The highest power of the unknown variable quantity is known as the degree of the polynomial equation, here the degree is 2. The solutions of a polynomial equation are those values of the variable quantity at which the value of the whole equation comes out to be 0. So, the two values obtained are known as the solutions of the given equation.