Question

How do you factor $ 2{x^2} + x - 3 = 0 $

Answer 1

Hint: Here in this question they have given the expression and it is an algebraic expression where it contains variables and constant terms. We have to find the factor $ 2{x^2} + x - 3 = 0 $ , so divide it by using variable and constant and hence find the solution.

Complete step-by-step answer:
The equation $ 2{x^2} + x - 3 = 0 $ is a quadratic equation and we get two factors on factorization. This equation is an algebraic equation which contains both constant term and variable term. To find the factor for the equation $ 2{x^2} + x - 3 = 0 $ we have a rule and it is a sum-product rule.
{Generally, the sum-product rule of the equation is
Consider the equation in general $ a{x^2} + bx + c = 0 $ where the product of a and c is written as sum of b. The numbers should satisfy the b by applying the sum rule. Here the sign convention plays an important role.}
Now consider the equation $ 2{x^2} + x - 3 = 0 $ , here a= 2 b=1 and c=-3
The product of ac is -6
The factors of -6 is 3 and -2 and the factors of -6 is -3 and 2
Now we will check which factors on the sum rule will satisfy the b
If we apply the sum rule to 3 and -2 the answer is 1
If we apply the sum rule to -3 and 2 the answer is -1
So we will consider 3 and -2 as factors of -6
Therefore the equation can be written as
 $ 2{x^2} + x - 3 = 0 $
 $ \Rightarrow 2{x^2} + 3x - 2x - 3 = 0 $
Rearrange the terms in the equation
 $ \Rightarrow 2{x^2} - 2x + 3x - 3 = 0 $
Let we take 2x as common in $ 2{x^2} - 2x $ and 3 as common in $ 3x - 3 $ and the equation is written as
 $ \Rightarrow 2x(x - 1) + 3(x - 1) = 0 $
Take (x-1) as common in the above equation we have
 $ (x - 1)(2x + 3) = 0 $
Therefore the factors of $ 2{x^2} + x - 3 = 0 $ is $ (x - 1)(2x + 3) = 0 $
So, the correct answer is “$ (x - 1)(2x + 3) = 0 $”.

Note: The sum product rule is used to find the factors and will obtain the solution for the question. In general the sum product is given as the equation in general $ a{x^2} + bx + c = 0 $ where the product of a and c is written as sum of b. The numbers should satisfy the b by applying the sum rule. Here the sign convention plays an important role. The sign conventions should be known to satisfy the equation