Question

How do you find the zeros of $3{{x}^{2}}-10x+3=0$ ?

Answer 1

Hint: Finding zeros of a polynomial is the same as finding factors of the polynomial. Zeroes are nothing but those numbers which when substituted in place of $x$would result in the entire expression equation to the right hand side of the equation which is a $0$. This process of finding out factors of the given polynomial or in this case quadratic equation is called factorization. It can be done in two ways. Either by splitting the middle term or by directly applying the quadratic equation form and find its roots. The quadratic equation formula is :
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ where $\sqrt{{{b}^{2}}-4ac}$ is called the discriminant of the of the quadratic equation.

Complete step by step solution:
Roots are the same as factors which are the same as zeroes of the given quadratic equation.
Here let us find the zeros by splitting the middle term.
Let us consider $f\left( x \right)=3{{x}^{2}}-10x+3=0$.
Let’s make use of a standard form of a quadratic equation to simplify things.
Let $h\left( x \right)=a{{x}^{2}}+bx+c$
Let’s factorize $f\left( x \right)$ by splitting the middle term.
We split the middle term, $bx$, in such a way so that we write it as the sum of two terms which when multiplied will give us the product of the first, $a{{x}^{2}}$, and last term,$c$, of the quadratic equation.
Upon comparing $f\left( x \right)$ with $h\left( x \right)$ , we conclude :
$\begin{align}
  & \Rightarrow a=3 \\
 & \Rightarrow b=-10 \\
 & \Rightarrow c=3 \\
\end{align}$
Now , let’s split the middle term :
$\begin{align}
  & \Rightarrow f\left( x \right)=3{{x}^{2}}-10x+3=0 \\
 & \Rightarrow f\left( x \right)=3{{x}^{2}}-1x-9x+3=0 \\
\end{align}$
Since $-1x-9x=-10x$ and $-1x\times -9x=9{{x}^{2}}$
$\begin{align}
  & \Rightarrow f\left( x \right)=3{{x}^{2}}-1x-9x+3=0 \\
 & \Rightarrow f\left( x \right)=x\left( 3x-1 \right)-3\left( 3x-1 \right)=0 \\
\end{align}$
Taking $3x-1$ common , we get the following :
$\Rightarrow f\left( x \right)=\left( 3x-1 \right)\left( x-3 \right)=0$
\[x=\dfrac{1}{3}\] or $x=3$are the zeroes of the given quadratic equation.

\[\therefore \] Hence, the zeroes of the quadratic equation $3{{x}^{2}}-10x+3=0$are $\dfrac{1}{3},3$.

Note: We can also try doing it using the formula. As we have already compared and found out the values of $a,b,c$ . We just have to plug it in the formula to get the values of $x$ which are roots or zeros to the given quadratic equation. We get the same answer both ways.