Question

How do you solve $3{{x}^{2}}-8x=0$ ?

Answer 1

Hint: We are given $3{{x}^{2}}-8x=0$, to solve this we learn about quadratic equation, number of solutions of quadratic equation. we will learn how to factor the quadratic equation, we will simplify by taking common terms out then we use zero product rule to get our answer. At the end we will also learn about quadratic formulas for solving such equations in an easy way and more speedy way.

Complete step by step answer:
We are given $3{{x}^{2}}-8x=0$, we are asked to solve the given problem.
First we observe that it has a maximum power of 2 so it is a quadratic equation.
Now we should know that a quadratic equation has 2 solutions or we say an equation of power ‘n’ will have an ‘n’ solution.
Now we have to solve the equation $3{{x}^{2}}-8x=0$to solve this equation we first take the greatest common factor possibly available to the terms.
As we can see that in $3{{x}^{2}}-8x=0$
First term is $3{{x}^{2}}$ which can be written as $3\times x\times x$ and the other term is $8x$ which can be written as $2\times 2\times 2\times x$
Now, we can see that both of the two terms' only common factor is ‘x’.
So we take it out, we get –
$3{{x}^{2}}-8x=x\left( 3x-8 \right)=0$
Now we use a zero product rule which says that if ‘a’ and ‘b’ products are zero then either one of the ‘a’ and ‘b’ must be zero.
So as we have $x\left( 3x-8 \right)=0$
So using above rule, we get –
Either $x=0$ or $3x-8=0$
So we solve them one by one.
$x=0$
Or
$\begin{align}
  & 3x-8=0 \\
 & 3x=8 \\
\end{align}$
We get by dividing both side
$x=\dfrac{8}{3}$
So, we get –
$x=0$ or $x=\dfrac{8}{3}$

So, we have that $x=0$ and $x=\dfrac{8}{3}$ as the solution of the above problem.

Note: There is another way by which we can solve this problem.
The other method is solving the quadratic equation $3{{x}^{2}}-8x=0$ using quadratic formula defined as –
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
In our equation $3{{x}^{2}}-8x=0$
We can see we have $a=3,b=-8\text{ and }c=0$
So, using $a=3,b=-8\text{ and }c=0$ in quadratic formula, we get –
$x=\dfrac{8\pm \sqrt{{{\left( -8 \right)}^{2}}}}{6}=\dfrac{8\pm 8}{6}$ as $\sqrt{{{\left( -8 \right)}^{2}}}=8$ we get –
$x=\dfrac{8+8}{6}\text{ and }x=\dfrac{8-8}{6}$
So,
$x=\dfrac{16}{6}\text{ and }x=\dfrac{0}{6}$
By simplifying, we get –
Solution are $x=\dfrac{8}{3}$ and $x=0$
Always remember that while picking a, b, or c we must be careful picking them with their sign, if we miss sign, we end up at the wrong solution.