Question

Find the roots of\[{x^2} - 4x - 8 = 0\] by the method of completing the square.

Answer 1

Hint: Quadratic equations are the equation that contains at least one squared variable which is equal to zero. Quadratic equations are useful in our daily life; they are used to calculate areas, speed of the objects, projection, etc. The quadratic equation is given as \[a{x^2} + bx + c = 0\]This is the basic equation which contains a squared variable \[x\]and three constants a, b and c.

Complete step-by-step answer:
Completing the square method is a technique for converting a quadratic polynomial of the form to the form of some value h and k.
In this question, roots are to be found by using completing square method where the constant term is kept on one side and the variables on the other side. Now the obtained equation is added with a constant term on both sides of the equation to make the variable a perfect square on one side of the equation.
For the given quadratic equation, \[{x^2} - 4x - 8 = 0\]
Bring the constant term on one side and variables on the other side of the equation, so that the given equation can be re-written as: \[{x^2} - 4x = 8\]
Now add a constant $ c = 4 $ on both sides of the equation to make LHS complete the square,
\[{x^2} - 4x + 4 = 8 + 4 - - - (i)\]
Equation (i) can be written as, \[{\left( {x - 2} \right)^2} = 12 - - - - (ii)\]
Now by solving the equation (ii) for the value of x,
\[
  {\left( {x - 2} \right)^2} = 12 \\
  x - 2 = \sqrt {12} \\
  x - 2 = \pm 2\sqrt 3 \\
  x = 2 \pm 2\sqrt 3 \\
 \]
Hence, we get the roots of a given quadratic equation\[{x^2} - 4x - 8 = 0\] as \[x = 2 + 2\sqrt 3 \] and \[x = 2 - 2\sqrt 3 \]

Note: In the quadratic equation if \[{b^2} - 4ac > 0\]the equation will have two real roots. If it is equal, \[{b^2} - 4ac = 0\]then the equation will have only one real root and when \[{b^2} - 4ac < 0\]then the root is in complex form.