Question

Solve the given quadratic equation by factorization method, \[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\]

Answer 1

Hint: We have to solve the following equation by the factorization method. In the factorization method, we have to break the middle term such that the sum of that form the middle term and the product of that term is the product of the first term and constant term. So, first, multiply the first term and constant part and then break the middle term according to the given condition.

Complete step-by-step solution:
Given:
A quadratic equation \[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\]
To find,
 The roots of the quadratic equation \[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\] by the factorization method.
Factorization method: In the factorization method, the middle term is stilted in two terms such that the product of those rems are the product of the first term and the constant term of the equation.
e.g. \[a{x^2} + bx + c = 0\] in this equation we have to split \[b\] such that the product of splatted terms is equal to the product of \[a\] and \[c\] .
So in this question the product of \[a\] and \[c\] are \[10\].
So now split \[7\] such that the product that number is \[10\]. The splitted numbers are \[2\] and \[5\].
\[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\]
The equation looks like after splitting is,
\[\sqrt 2 {x^2} + 2x + 5x + 5\sqrt 2 = 0\]
Taking the common terms outside,
\[\sqrt 2 x(x + \sqrt 2 ) + 5(x + \sqrt 2 ) = 0\]
After further arranging.
\[(\sqrt 2 x + 5)(x + \sqrt 2 ) = 0\]
From here, we get two values of \[x\] that are the roots of the equation \[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\] is
\[ \Rightarrow x = \dfrac{{ - 5}}{{\sqrt 2 }}\]
\[ \Rightarrow x = - \sqrt 2 \]
Final answer:
the roots of the equation \[\sqrt 2 {x^2} + 7x + 5\sqrt 2 = 0\] are
 \[ \Rightarrow x = \dfrac{{ - 5}}{{\sqrt 2 }}\] and
\[ \Rightarrow x = - \sqrt 2 \]

Note: To solve this type of question we have to use the factorization method to find the roots of the equation. In the factorization method, you may make mistakes while splitting the middle terms and taking common ones from all the terms. After taking common terms from the terms we will get two terms from that also we have to take common terms and equate both of them to \[0\].