Question

What are the solution(s) of: $3{{x}^{2}}-8x+5=0$?

Answer 1

Hint: Use the middle term split method to factorize the quadratic equation. Split -8x into two terms such that their sum equals -8x and the product equals \[15{{x}^{2}}\]. For this process, find the prime factors of 15 and combine them in a suitable way such that the conditions are satisfied. Finally, write the equation as the product of two linear binomials and equate them with 0 to find the values of x.

Complete step-by-step solution:
Here we have been asked to find the solution(s) of the quadratic equation $3{{x}^{2}}-8x+5=0$. Here we will use the middle term split method to get the two values of x. $\because 3{{x}^{2}}-8x+5=0$
First we need to factorize the above expression. Let us use the middle term split method for the factorization. In this method we have to split the middle term which is -8x into two terms such that their sum equals -8x and the product is equal to the product of constant term (5) and \[3{{x}^{2}}\], i.e. \[15{{x}^{2}}\]. To do this, first we need to find all the prime factors of 15.
We can write \[15=3\times 5\] as the product of its primes. Now, we have to group these factors such that our conditions of the middle terms split method are satisfied. So we have,
(i) \[\left( -3x \right)+\left( -5x \right)=8x\]
(ii) \[\left( -3x \right)\times \left( -5x \right)=15{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic equation can be written as: -
\[\begin{align}
  & \Rightarrow 3{{x}^{2}}-3x-5x+5=0 \\
 & \Rightarrow 3x\left( x-1 \right)-5\left( x-1 \right)=0 \\
\end{align}\]
Taking (x – 1) common we get,
\[\Rightarrow \left( x-1 \right)\left( 3x-5 \right)=0\]
Substituting each term equal to 0 we get,
\[\Rightarrow \left( x-1 \right)=0\] or \[\left( 3x-5 \right)=0\]
\[\Rightarrow x=1\] or \[x=\dfrac{5}{3}\]
Hence, the above two values of x are the solutions of the quadratic equation.
Note: Note that the solution (s) of a quadratic equation is also known as its roots. You can also use the discriminant method to solve the question. What you have to do is, apply the quadratic formula given as $y=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ and simplify it to get the two roots as answers. Here, a is the coefficient of ${{x}^{2}}$, b is the coefficient of x and c is the constant term.