Question

Factorize the following \[3{x^2} - 10x + 8\]
\[\left( A \right){\text{ }}\left( {3x - 4} \right)\left( {x - 2} \right)\]
\[\left( B \right){\text{ }}\left( {3x + 4} \right)\left( {x - 2} \right)\]
\[\left( C \right){\text{ }}\left( {3x - 4} \right)\left( {x + 2} \right)\]
\[\left( D \right)\] None of these

Answer 1

Hint: The standard form of the quadratic equation is \[a{x^2} + bx + c = 0\] , compare the equation given in the question with the standard form of the equation to get the values of a, b and c. Then find two numbers which on multiplication gives the product of a and c , and the sum of which is b. After finding those numbers, rewrite the middle of the equation with those numbers. Then factor the first two and last two terms separately to find the factors of the given equation.

Complete step by step answer:
The equation given to us is
\[3{x^2} - 10x + 8 = 0\] ------------ (i)
And the standard form of the quadratic equation is
\[a{x^2} + bx + c = 0\] ------------ (ii)
Where a, b and c can have any value except that a can’t be zero.
Now we will compare the equations (i) and (ii) to get the values of a, b and c. Therefore on comparing we get
\[a = 3\]
\[b = - 10\]
And \[c = 8\]
The given equation is \[3{x^2} - 10x + 8 = 0\] . Now we will find the two numbers that multiply to give ‘ac’ and on adding gives b. That is we have to find two numbers which on multiplying gives \[24\] and on adding gives \[ - 10\] .
As on multiplying \[ - 6\] and \[ - 4\] we get \[24\] and the sum of \[ - 6\] and \[ - 4\] is \[ - 10\] . Therefore, in this case the required pair is \[\left( { - 6, - 4} \right)\]
Now replace \[ - 10x\] by \[ - 6x\] and \[ - 4x\] in the equation (i). By doing this the equation (i) becomes
\[3{x^2} - 6x - 4x + 8 = 0\]
Now factor the first two and the last two terms separately
\[3x\left( {x - 2} \right) - 4\left( {x - 2} \right) = 0\]
So,
\[\left( {3x - 4} \right)\left( {x - 2} \right) = 0\]
Therefore the required factors of the given equation \[3{x^2} - 10x + 8 = 0\] are \[\left( {3x - 4} \right)\left( {x - 2} \right)\] .

So, the correct answer is “Option A”.

Note:
Keep in mind the standard form of the quadratic equation, \[a{x^2} + bx + c = 0\] . Remember that here the value of ‘a’ can’t be zero, it is greater than or equal to one. Remember the procedure of factorization of a quadratic equation. Do the calculations carefully while finding the factors of the equation.