Question

How do you solve \[9{{x}^{2}}=324\]

Answer 1

Hint: Here we have to solve the equation with the help of quadratic equation.
We have $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Where $a,b$ and $c$ are the co-efficient.

Complete step by step solution:
In the above equation we have to find out the value of $x.$
So, first rearrange the equation as follow,
$9{{x}^{2}}-324=0...(i)$
The quadratic equation is as $9{{x}^{2}}+bx+c=0$ Where $a,b$ and $c$ are co-coefficient of the equation.
Now, equation with equation $(i)$we get.
$a=9,$ here the value of $a$ is $9.$
$b=0$, here the value of $b$ is $0.$
$c=-324,$ here the value of $c$ is $-324$
Now, substituting the value in quadratic form as,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
We get,
$x=\dfrac{0\pm \sqrt{{{0}^{2}}-4\times 9\times \left( -324 \right)}}{2\times 9}$
Further solving we get,
$x=\dfrac{\pm \sqrt{-\left( 4\times 9\times \left( -324 \right) \right)}}{18}$
Now multiplying $4$ with $9$ and $-324$, we have $x=\dfrac{\pm \sqrt{11664}}{18}$
Now dominating root, we get.
$x=\dfrac{\pm 108}{18}$
Here we have two value for $x$ we get,
$x=\dfrac{+108}{18}$ and $x=\dfrac{-108}{18}$

Here we have two value for $x$ we get,$x=+6$ and $x=-6$

Additional Information:
When we move any mathematical expression from left to right side or vice versa then the sign of the expression gets reversed.
Like, $2x+1=2,$ if we move $1$ from left side to right side i.e. after equal to then the positive sign of $+1$ gets converted into negative sign.
Thus, it will equal to $2x=2-1$
Similarly, in $2-3x=-4,$ If here we more $-3x$ from left side to right side then it will become positive, and if we move $-4$ from right side to left side it will become $+4.$
So, $2-3x=-4\Rightarrow 3x=2+4$
Similarly, if $2x=4$ then here $2$ is in multiplication with $x,$ in order to determine the value of $x$ we have to replace $2$ from left side to right side, so it will become divided.
i.e. $2x=4\Rightarrow x=\dfrac{4}{2}$ here, $2$ which are in multiplication on the left side, when transferred to the right side, will be converted into a divide.
In the same way, if $x=5,$ here $2$ is in division with $x$ on the left side, so when we solve the equation then it will be transferred to the right side, and converted into multiplication.
Like, $\dfrac{1}{2}x=5\Rightarrow x\left( 5\times 2 \right)$
There are two ways to solve the equation of linear equation,
(1) By separating the like terms, like terms are those numbers which are similar in nature, like $\left( 2x,\dfrac{1}{2}x,3x \right)$ or any constant.
(2) By adding or subtracting or by doing arithmetic processes. Like if we have to solve.
$2x\times 3=11$
Here, as we have to determine the value of $2x,$ As $3$ is in addition with $2x$ in left side,
So, in order to neutralise it. Will subtract $3$ from side,
So, equation becomes,
$2x+3-3=11-3$
As, $2x=8\Rightarrow x=\dfrac{8}{2}=4$

Note: While transferring the digits or constants or any variables or number from left hand side to right hand side, make sure you are reversing its symbol.
For any mathematical operation, always follow only the BODMAS rule.