Question

How do you solve \[(2x + 3)\ (3x + 7)\ = 0\] by using the factoring method ?

Answer 1

Hint: In this question, we need to solve the given expression and need to find the roots of the given expression by using the factoring method . An algebraic expression is nothing but it is built up with integers, constants, variables and mathematical operations (addition, subtraction, multiplication, division etc… ) In mathematics, a symbol (letter) which doesn’t have a value is called a variable. Similarly when it has a fixed value it is called constant .Here we need to split the given expression and equate the expression to \[0\]. Then we can easily find the roots of the given expression.

Complete step-by-step solution:
Given,
\[(2x + 3)\ (3x + 7)\ = 0\]
In the given expression, the product of two simpler linear expressions is equal to \[0\] . Thus we can find the value of \[x\] by splitting each expression to \[0\] .
\[\Rightarrow (2x + 3) = 0\] and \[(3x – 7) = 0\]
On simplifying,
We get,
\[\Rightarrow \ 2x = - 3\] and \[3x = 7\]
Thus we get,
\[\Rightarrow \ x = - \dfrac{3}{2}\] and \[x = \dfrac{7}{3}\]
Thus, \[x = - \dfrac{3}{2}\] and \[x = \dfrac{7}{3}\] are two roots of the equation \[(2x + 3)(3x – 7) = 0\]
Final answer :
The two roots of the equation \[(2x + 3)(3x – 7) = 0\] are \[- \dfrac{3}{2}\] and \[\dfrac{7}{3}\].

Note: An algebraic expression is nothing but it is a product of two simpler linear expressions. The concept used in this question to solve the given expression is solutions of quadratic equations by Factorization . Factorization is nothing but writing a whole number into smaller numbers of the same kind. By using algebraic formulas and also by taking the common terms outside, we can factorize the given expression. In other words, factorization is known as the decomposition of the mathematical objects to the product of smaller objects. Matrices also possess the process of factorization. The formula of factorization is
\[N = X^{a}Y^{b}Z^{c}\]
Where \[a,\ b,\ c\] are the exponential powers of the factor. There are five methods in factorization. We can reduce any algebraic expressions into smaller objects where the equations are represented as the product of factors.