Question

Solve by completing square method:
\[{x^2} + 8x + 5\].

Answer 1

Hint: Hint: While solving the quadratic equation by completing square method, the third term is \[{\left( {\dfrac{1}{2} \times {\text{ coefficient of x}}} \right)^2}\]
The equation is rearranged so that the third term is placed to the right hand side of the equation.

Complete step by step solution:
To obtain the roots of a quadratic equation, you can use the completing square method. A quadratic equation is the polynomial equation of degree 2.
The term ‘Quad’ has a meaning of ‘four’, whereas the term ‘Quadratic’ has a meaning of ‘to make square’.
The following approach is used while solving the quadratic equation by completing the square method.
The given quadratic equation is \[{x^2} + 8x + 5 = 0\].

The third term is \[{\left( {\dfrac{1}{2} \times {\text{ coefficient of x}}} \right)^2} = {\left( {\dfrac{1}{2} \times {\text{ 8}}} \right)^2} = 16\]

Add 16 to both sides of quadratic equation:
\[{x^2} + 8x + 16 + 5 = 16\]
Rearrange above equation by placing 5 to the right hand side of the equation
\[{x^2} + 8x + 16 = 16 - 5 = 11\]… …(1)
But \[{x^2} + 8x + 16\] is equal to \[{\left( {x + 4} \right)^2}\]
Hence, the equation (1) becomes
\[{\left( {x + 4} \right)^2} = 11\]
Take square root on both sides of the equation:
\[x + 4 = \sqrt {11} \] or \[x = - \sqrt {11} - 4\]
These are the roots of the quadratic equation

Note: The standard form of a quadratic equation is \[a{x^2} + bx + c = 0\].
Here, x is the variable and a,b and c are the real numbers.
The roots of the quadratic equation can be given by the following equation.
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]