Question

How do you solve $ 3{{x}^{2}}-10x+3=0 $ ?

Answer 1

Hint: In this question, we have to find the value of x. The equation given in the problem is in the form of the quadratic equation $ a{{x}^{2}}+bx+c=0 $ . So, we will apply splitting the middle method, to get two values of x. First, we will split the middle term of the equation is the sum of $ -9x $ and $ -x $ . After that, we will take common 3x in the first two terms and -1 in the last two terms. After the necessary calculations, we get two values of x, which is our required solution.

Complete step by step answer:
According to the question, we have to find the value of x.
Thus, to solve this problem we will use the splitting the middle terms method.
The equation given to us is $ 3{{x}^{2}}-10x+3=0 $ ---------- (1)
As we see know, the equation (1) is in the general form of the quadratic equation $ a{{x}^{2}}+bx+c=0 $, thus on comparing both the equations, we get
 $ a=3\text{, }b=-10\text{, and }c=3 $
Therefore, we will apply to split the middle term method, that is, we will express the middle term of an equation $ b $ in such a way that it will be the sum of the factors of $ a.c $.
So, we see that $ ac=3.(3)=9 $ , that is
 $ 9=(-9).(-1) $ and $ -9-1=-10 $
So, we will rewrite the middle term as a sum of -9x and -1x, we get
 $ \Rightarrow 3{{x}^{2}}-9x-1x+3=0 $
Therefore, we take 3x common in the first two terms and -1 common in the last two terms, we get
  $ \Rightarrow 3x(x-3)-1(x-3)=0 $
Now, we take common (x-3) in the above equation, we get
 $ \Rightarrow (x-3)(3x-1)=0 $
So, either $ x-3=0 $ ----------- (3) or
 $ 3x-1=0 $ --------- (4)
Thus, first, we will solve equation (3), that is
 $ \Rightarrow x-3=0 $
Now, we will add 3 on both sides in the above equation, we get
 $ \Rightarrow x-3+3=0+3 $
As we know, the same terms with opposite signs cancel out, therefore, we get
 $ \Rightarrow x=3 $
Now, we will solve equation (4), which is
 $ 3x-1=0 $
Now, we will add 1 on both sides in the above equation, we get
 $ \Rightarrow 3x-1+1=0+1 $
As we know, the same terms with opposite signs cancel out, we get
 $ \Rightarrow 3x=1 $
Now, we will divide 3 on both sides of the equation, we get
 $ \Rightarrow \dfrac{3}{3}x=\dfrac{1}{3} $
 $ \Rightarrow x=\dfrac{1}{3} $
Therefore, for the equation $ 3{{x}^{2}}-10x+3=0 $ , the value of x is $ 3 $ and $ \dfrac{1}{3} $ .

Note:
While solving this problem, do mention all the steps properly to avoid confusion and mistakes. One of the alternative methods to solve this problem is to use the discriminant formula $ x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ to get the value of x, which is our required solution to the problem.