Question

How do you simplify the product $(6x – 5) (3x + 1)$ and write it in the standard form.

Answer 1

Hint: We will first use the property that $(a + b) (c + d) = a (c + d) + b (c + d)$. Now, we will use the distributive property to further simplify them and at last we will club the like terms.

Complete step-by-step solution:
We are given that we are required to foil $(6x – 5) (3x + 1)$.
We know that we have a property given by the following expression:-
$ \Rightarrow $$(a + b) (c + d) = a (c + d) + b (c + d)$
Now, replacing $a$ by $6x$, $b$ by $– 5$, $c$ by $3x$ and $d$ by $1$, we will then obtain the following equation:-
$ \Rightarrow $$(6x – 5) (3x + 1) = 6x (3x + 1) - 5 (3x + 1)$
Term the above equation as equation number $1$.
So, we have $(6x – 5) (3x + 1) = 6x (3x + 1) - 5 (3x + 1)$ …………….(1)
Now, we will first use the distributive property in $6x (3x + 1)4:
$ \Rightarrow $46x (3x + 1) = (6x) (3x) + (6x) (1)$
Simplifying the brackets in the right hand side, we will then obtain the following equation:-
$ \Rightarrow 6x(3x + 1) = 18{x^2} + 6x$ ……………..(2)
Now, we will use the distributive property in - 5 (3x + 1):
$ \Rightarrow $$- 5 (3x + 1) = (- 5) (3x) + (- 5) (1)$
Simplifying the brackets in the right hand side, we will then obtain the following equation:-
$ \Rightarrow $$- 5 (3x + 1) = - 15 x - 5$ ……………..(3)
Putting the equation numbers (2) and (3) in equation number (1), we will then obtain the following equation:-
$ \Rightarrow 6x(3x + 1) - 5(3x + 1) = 18{x^2} + 6x - 15x - 5$
We can write the above equation as follows:-
$ \Rightarrow (6x - 5)(3x + 1) = 18{x^2} + 6x - 15x - 5$
Clubbing the like terms in the right hand side, we will then obtain the following equation:-
$ \Rightarrow (6x - 5)(3x + 1) = 18{x^2} - 9x - 5$

Therefore $18{x^2} - 9x - 5$ is the required answer.

Note: The students must know the definition of the distributive property which we have used in the solution of the above question:-
Distributive Property: It states that for any three numbers a, b and c, we have the following equation with us: $ \Rightarrow $a (b + c) = ab + ac
The students must also note that the standard form of an equation is written when we write the term with highest power to lower power in the same order.