Question

How do you solve \[10{{x}^{2}}-15x=0\]?

Answer 1

Hint: The equation \[10{{x}^{2}}-15x=0\], given in the above question is a quadratic equation. SO it will have two roots. It can be simplified by dividing it by the factor $5$ which is common to the LHS terms to get the simplified equation as $2{{x}^{2}}-3x=0$. Now, the factor $x$ has to be taken outside so that the LHS will become factorized from which the roots can be easily determined using the zero product rule.

Complete step by step solution:
The equation given in the question is
$\Rightarrow 10{{x}^{2}}-15x=0$
Dividing both sides of the above equation by $5$ we get
$\begin{align}
  & \Rightarrow \dfrac{10}{5}{{x}^{2}}-\dfrac{15}{5}x=\dfrac{0}{5} \\
 & \Rightarrow 2{{x}^{2}}-3x=0 \\
\end{align}$
In the above equation we can see that the factor $x$ is common. Taking it outside, we get
$\Rightarrow x\left( 2x-3 \right)=0$
Now, by the zero product rule we know that if $a\cdot b=0$, then either $a=0$ or $b=0$. THereofre, from the above equation we can write
$\Rightarrow x=0$
And
$\Rightarrow 2x-3=0$
Adding $3$ both sides
$\begin{align}
  & \Rightarrow 2x-3+3=0+3 \\
 & \Rightarrow 2x=3 \\
\end{align}$
Finally, dividing both sides by $2$ we get
$\begin{align}
  & \Rightarrow \dfrac{2x}{2}=\dfrac{3}{2} \\
 & \Rightarrow x=\dfrac{3}{2} \\
\end{align}$

Hence, the solutions of the given equation are $x=0$ and $x=\dfrac{3}{2}$.

Note: In the LHS of the given equation, the factor $x$ is common. Do not divide the given equation by this factor, since this factor is a variable and the division by a variable is not allowed. We can only divide an equation by the constant factors, just like we divided the given equation by the factor $5$ which is a constant. In the above solution, we have used the factorization method to solve this question. Other methods such as middle term splitting and the quadratic formula are also employable. But since the constant term is missing in the given quadratic equation, the factorization method is the easiest.