Question

Find the factor of \[{x^2} + 3\sqrt 2 x + 4\].

Answer 1

Hint: In this problem, we need to use the middle term splitter formula to find the factor of given quadratic expression. We need to split the middle term in such a way that the sum is equal to the \[x\] coefficient and the product is equal to the last term.

Complete step by step answer:
While doing the factor of a quadratic expression of the form \[a{x^2} + bx + c\] using middle term split method, we need to split the middle term in such a way that the sum is equal to \[b\] and product is equal to \[ac\].
The given quadratic expression is \[{x^2} + 3\sqrt 2 x + 4\].
Now, we need to split the middle term in such a way that the sum is equal to \[3\sqrt 2 \] and the product is equal to 4. The proper splitting of the middle term is shown below.
\[3\sqrt 2 = 2\sqrt 2 + \sqrt 2\]
Now, the factor of the given quadratic expression \[{x^2} + 3\sqrt 2 x + 4\] is calculated as shown below.
\[
  \,\,\,\,\,\,{x^2} + 3\sqrt 2 x + 4 \\
   \Rightarrow {x^2} + \left( {2\sqrt 2 + \sqrt 2 } \right)x + 4 \\
   \Rightarrow {x^2} + 2\sqrt 2 x + \sqrt 2 x + 4 \\
   \Rightarrow x\left( {x + 2\sqrt 2 } \right) + \sqrt 2 \left( {x + 2\sqrt 2 } \right) \\
   \Rightarrow \left( {x + \sqrt 2 } \right)\left( {x + 2\sqrt 2 } \right) \\
\]

Thus, the factor of the quadratic expression \[{x^2} + 3\sqrt 2 x + 4\] is \[\left( {x + \sqrt 2 } \right)\left( {x + 2\sqrt 2 } \right)\].

Note: The quadratic equation is a function that can be described by an equation of the form \[f\left( x \right) = a{x^2} + bx + c\], where \[a\] cannot be equal to zero. We can solve the quadratic equation using the Shridhar Acharya formula as shown below.
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]