Question

Factorize: $2{a^2} + bc - 2ab - ac$

Answer 1

Hint: In this question, we are given an algebraic expression containing three alphabets and numbers related to each other by some arithmetic operations. The alphabets a, b and c represent some unknown variable quantities. Out of these three unknown variables, “a” is raised to the power 2, so it is a polynomial equation in terms of a, and terms containing “b” and “c” act as the coefficients and the constants. As the highest power of “a” in this equation is 2, so the polynomial equation has a degree 2, and is thus a quadratic equation. We will factorize the given equation after converting it to the standard quadratic form.

Complete step-by-step solution:
The standard form of a quadratic equation is $a{x^2} + bx + c$ , the given equation in the standard form will be $2{a^2} - 2ab - ac + bc$ or $2{a^2} + ( - 2b + c)a + bc$ . We can factorize this equation by taking the common terms out and then simplifying the expression.
$
   \Rightarrow 2{a^2} + bc - 2ab - ac = a(2a - c) - b(2a - c) \\
   \Rightarrow 2{a^2} + bc - 2ab - ac = (a - b)(2a - c) \\
 $
Hence, the factored form of $2{a^2} + bc - 2ab - ac$ is $(a - b)(2a - c)$ .


Note: Factors of an equation are simply the expressions that completely divide the given equation. The standard form of a quadratic equation is $a{x^2} + bx + c = 0$ . To find the factors of the given equation, we get the values of a, b and c by comparing the given equation with the standard form. Then we will try to write b as a sum of two numbers such that their product is equal to the product of a and c, that is, ${b_1} \times {b_2} = a \times c$ , this method is known as factorization. We find the value of ${b_1}$ and ${b_2}$ by hit and trial. By putting the obtained equation equal to zero, we can find the value of “a” in terms of “b” and “c”.