Question

How do you factor $5{x^2} + 2x - 3 = 0$?

Answer 1

Hint: According to the question we have to determine the factors of the given quadratic expression which is $5{x^2} + 2x - 3 = 0$. So, basically we have to determine the roots or the zeroes of the quadratic expression which will satisfy the given equation.
First of all we have to find the L.C.M. of the constant term and the coefficient of ${x^2}$ to obtain the coefficient of x which is as given in the quadratic expression by adding or subtracting the factors are obtained after the L.C.M.
Now, we have to open all the brackets and then solve the expression obtained by multiplying, adding and subtracting the terms which can be.
Now, to obtain the roots or zeros we have to compare them with the right hand side which is 0 to obtain the required value of the variable which is x according to the given expression.

Complete step-by-step solution:
Step 1: First of all we have to find the L.C.M. of the constant term and the coefficient of ${x^2}$ which is as mentioned in the solution hint. Hence, factors of the terms,
$ = 3,5$
Step 2: Now, we have to obtain the coefficient of x which is as given in the quadratic expression by adding or subtracting the factors obtained after the L.C.M. as we have already obtained in the solution step 1. Hence,
$
   \Rightarrow 5{x^2} + (5 - 3)x - 3 = 0 \\
   \Rightarrow 5{x^2} + 5x - 3x - 3 = 0
 $
Step 3: Now, we have to solve the expression obtained just above to obtain the roots,
$
   \Rightarrow 5x(x + 1) - 3(x + 1) = 0 \\
   \Rightarrow (x + 1)(5x - 3) = 0
 $
Step 4: Now, to obtain the roots or zeros we have to compare them with the right hand side which is 0 to obtain the required value of the variable which is x according to the given expression. Hence,
\[
   \Rightarrow (x + 1) = 0 \\
   \Rightarrow x = - 1
 \]
And,
\[
   \Rightarrow (5x - 3) = 0 \\
   \Rightarrow x = \dfrac{3}{5}
 \]

Hence, we have determined both of the zeroes or the factor of the given quadratic expression which are \[x = - 1\] and \[x = \dfrac{3}{5}\].

Note: On solving a quadratic expression we know that there are two possible roots or we can say that there can be two zeroes.
On substituting both of the roots obtained will satisfy the given quadratic expression means on substituting the roots obtained in the quadratic expression it will become zero or give zero as a result.