Question

How do you simplify $(3x + 2)(2x - 1)$?

Answer 1

Hint: To simplify the function, one must open the brackets applying the distributive law. Hence, the first term of the first bracket is multiplied with both terms of the second bracket and the same applies for the second term of the first bracket. Hence every term of one bracket must be multiplied with every other term of the second bracket.

Formula used: Product of a sum and a difference: $(a + b)(c - d) = ac - ad + bc - bd$

Complete step-by-step solution:
Here, we are given a sum and a difference of two terms and as there is no operation shown between the two brackets, we have to multiply the sum and the difference.
We have the distributive law which can be expressed as,
$a(b + c) = ab + ac$
The distributive law explains the multiplication of a number to a sum or difference of two numbers.
But, to understand the distributive law for the multiplication of a sum of two numbers and a difference of two numbers, we can assume either the sum or the difference as a single value.
Hence, let’s assume the first term as a single value
$3x + 2 = a$
Let’s consider this equation as Equation $(1)$
Substituting this assumption in the function
$\Rightarrow a(2x - 1)$
Now, applying the distributive law,
$\Rightarrow a \times 2x - a \times 1$
Now, substituting the value of $a$ from Equation $(1)$ ,
$\Rightarrow (3x + 2) \times 2x - (3x + 2) \times 1$
Now, applying again the distributive law to both terms,
$\Rightarrow 3x \times 2x + 2 \times 2x - 3x \times 1 - 2 \times 1$
Hence, now every term of the first bracket of the given function is multiplied with every term of the second bracket
We know that multiplying a value or variable with itself adds the power of $\;2$ to the value or variable.
Hence simplifying the multiplication of terms, we get
$\Rightarrow 6{x^2} + 4x - 3x - 2$
Now, taking the difference of terms with the same power of $x$ ,
$\Rightarrow 6{x^2} + x - 2$

Hence, $6{x^2} + x - 2$ is the simplified function of the given function.

Note: Here, as we assumed the first term as a single value, we can also assume the second bracket as a single value. The rule is to multiply the terms of one bracket with every other term of the other bracket. This rule can also be applied to the multiplication of two sums or two differences.