Question

Solve equation $\dfrac{9}{2}x = 5 + {x^2}$ using factorization method.

Answer 1

Hint: The word quadratic means second-degree values of the given variables.
Then by further simplifying, the quadratic equation $a{x^2} + bx + c = 0$ can be written in the form of $(x - \alpha )(x - \beta ) = 0$.
In this equation, the factors of the quadratic equation are$(x - \alpha )$ and $(x - \beta )$.
So, to determine the roots of the equation, these factors of the equation will be made equal to $0$.
That is,
$(x - \alpha ) = 0$ and $(x - \beta ) = 0$
Which gives,
$x = \alpha $ and $x = \beta $
So, $\alpha $and $\beta $ are the roots of the quadratic equation in general form.
Here we are asked to solve the given quadratic equation that is we have to find its roots, since it is an equation of order two it will have two roots. First, we will make the given equation as factors of its roots that are in the form $(x - \alpha )(x - \beta ) = 0$ then we will find the values of $\alpha $and $\beta $ .

Complete step by step solution:
It is given that $\dfrac{9}{2}x = 5 + {x^2}$ we aim to solve this equation, that is we have to find its roots.
We know that the number of roots of an equation is equal to its degree. Here the degree of the given equation is two so we will get two roots for the given equation.
Given that $\dfrac{9}{2}x = 5 + {x^2}$. Now multiply with the number $2$ on both sides then we get
$\dfrac{{9 \times 2}}{2}x = 2(5 + {x^2}) $
$\Rightarrow 9x = 10 + 2{x^2}$
And thus we get
$2{x^2} - 9x + 10 = 0$
Now convert the value
$ - 9x = - 4x - 5x$
Then we get $2{x^2} - 4x - 5x + 10 = 0$
Which can also be written as $2{x^2} - 9x + 10 = 0$
Hence taking the common values out from
$2{x^2} - 4x - 5x + 10 = 0$
we get $2x(x - 2) - 5(x - 2) = 0$
The factors of the quadratic equation are $(x - \alpha )$ and $(x - \beta )$. And thus, we get $2x(x - 2) - 5(x - 2) = 0$
$ \Rightarrow (2x - 5)(x - 2) = 0$
thus $2x - 5 = 0$ and $x - 2 = 0$
Hence, we get $x = \dfrac{5}{2}$ and $x = 2$ as the roots of the given problem.

Note: The quadratic equations can also be solved by using the formula method. Let \[a{x^2} + bx + c = 0\] be a quadratic equation then the roots of this equation are given by \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. In the above problem we got a quadratic equation after simplifying the given equation that is $2{x^2} - 9x + 10 = 0$ here $a = 2,b = - 9$ and $c = 10$ by substituting these in the formula and simplifying it will give us the roots of the given equation.