Question

Factorize \[{x^2} + 7\sqrt 6 x + 60\]

Answer 1

Hint: We compare the given quadratic equation with general quadratic equation and write the values of coefficients. Break the coefficient of ‘x’ in such a way that their sum is equal to coefficient of ‘x’ and their product is equal to the product of coefficient of \[{x^2}\] and the constant term.
* General form of a quadratic equation is \[a{x^2} + bx + c = 0\]
* Factorization method: If ‘p’ and ‘q’ are the roots of a quadratic equation, then we can say the quadratic equation is \[(x - p)(x - q) = 0\]

Complete step-by-step solution:
We are given the equation \[{x^2} + 7\sqrt 6 x + 60\]..............… (1)
Since we know general form of a quadratic equation is \[a{x^2} + bx + c = 0\]
On comparing equation (1) with general equation we can write \[a = 1,b = 7\sqrt 6 ,c = 60\]
Now we calculate the product of coefficient of \[{x^2}\] and the constant term
Since coefficient of \[{x^2}\] is 1 and the constant term is 60
Then the value of product is \[60 \times 1 = 60\]............… (2)
We have to break the coefficient of ‘x’ i.e. \[7\sqrt 6 \] in such a way that the sum of those terms give \[7\sqrt 6 \] and the product of those terms gives 60.
We write prime factorization of the number 60
\[ \Rightarrow 60 = 2 \times 2 \times 3 \times 5\]
So we can form \[60 = 5 \times 2 \times 6\], but here we are dealing with under root of 6, so we will take both the terms as multiples of \[\sqrt 6 \] so that on multiplication they give an answer 6.
We can write \[7\sqrt 6 = 5\sqrt 6 + 2\sqrt 6 \]
Substitute the value of \[7\sqrt 6 = 5\sqrt 6 + 2\sqrt 6 \] in equation (1)
\[ \Rightarrow {x^2} + 7\sqrt 6 x + 60 = {x^2} + \left( {5\sqrt 6 + 2\sqrt 6 } \right)x + 60\]
Open the bracket in RHS of the equation
\[ \Rightarrow {x^2} + 7\sqrt 6 x + 60 = {x^2} + 5\sqrt 6 x + 2\sqrt 6 x + 60\]
Take ‘x’ common from first two terms and \[2\sqrt 6 \] common from last two terms in RHS
\[ \Rightarrow {x^2} + 7\sqrt 6 x + 60 = x\left( {x + 5\sqrt 6 } \right) + 2\sqrt 6 \left( {x + 5\sqrt 6 } \right)\]
Take \[\left( {x + 5\sqrt 6 } \right)\] common and pair the remaining factors in RHS
\[ \Rightarrow {x^2} + 7\sqrt 6 x + 60 = \left( {x + 5\sqrt 6 } \right)\left( {x + 2\sqrt 6 } \right)\]

\[\therefore \]Factorization of \[{x^2} + 7\sqrt 6 x + 60\] is \[\left( {x + 5\sqrt 6 } \right)\left( {x + 2\sqrt 6 } \right)\]

Note: Students many times make mistake while breaking the coefficient of ‘x’ as they write under root value as one of the term and other term as something else, keep in mind two numbers can only be added if they are similar i.e. two numbers that are multiples of same under root value can be added together.