Question

How do you simplify $3{y^2} - 2y?$

Answer 1

This question describes the operation of addition/ subtraction/ multiplication/ division. The final answer would be a simplified form of a given question. This question can be solved by using a quadratic equation and we can replace the term $x$ with $y$. This question can also be solved by finding the greatest common factor.

Complete step by step solution:
The given problem is shown below,
$3{y^2} - 2y?$
This equation can also be written as,
$3{y^2} - 2y = 0 \to \left( 1 \right)$
We know that,
The basic form of a quadratic equation is, $a{x^2} + bx + c = 0 \to \left( 2 \right)$(Here, $x$ is replaced with$y$)
So, we get
$y = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \to \left( 3 \right)$
By comparing the equation$\left( 1 \right)$and$\left( 2 \right)$, we get
$\left( 1 \right) \to 3{y^2} - 2y = 0$
$\left( 2 \right) \to a{x^2} + bx + c = 0$
So, we get
$a = 3,b = - 2$and$c = 0$.
So, the equation$\left( 3 \right)$becomes
$\left( 3 \right) \to y = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

$$y = \dfrac{{ - \left( { - 2} \right) \pm \sqrt {{{\left( { - 2} \right)}^2} - 4 \times 3 \times 0} }}{{2 \times 3}} \\\ y = \dfrac{{2 \pm \sqrt 4 }}{6} \\\$$

So, we get

$$y = \dfrac{{2 \pm 2}}{6} \\\ y = \dfrac{{2\left( { - 1 \pm 1} \right)}}{6} \\\$$

$y = \dfrac{{1 \pm 1}}{3}$
Case: 1

$$y = \dfrac{{1 + 1}}{3} \\\ y = \dfrac{2}{3} \\\$$

Case: 2

$$y = \dfrac{{1 - 1}}{3} \\\ y = 0 \\\$$

By using the values of $y$, we can write the following equation

$$3y - 2 = 0 \\\ y = 0 \\\$$

The above equations can also be written as,
$\left( {3y - 2} \right).y = 0$
When the question is $3{y^2} - 2y = 0$, the answer becomes $\left( {3y - 2} \right).y = 0$. So when the question is $3{y^2} - 2y$, the answer becomes $\left( {3y - 2} \right).y$.
So, the final answer is
$3{y^2} - 2y = \left( {3y - 2} \right).y$

Note: This type of question involves the operation of addition/ subtraction/ multiplication/ division. This question can also be solved by finding the greatest common factor. In the given question we have $3{y^2} - 2y$, the greatest common factor of the given equation is $y$. So, we can take $y$ it as a common term. So, the given question can also be written as $\left( {3y - 2} \right).y$. By this method, we can easily solve these types of questions. Note that we shouldn’t take $1$ as the greatest common factor to solve the question. If we take $1$ as a greatest common factor we won’t get a simplified form of the given equation.