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Question

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

How do you solve 9x2=3249{{x}^{2}}=324

Explanation

Solution

Here we have to solve the equation with the help of quadratic equation.
We have x=b±b24ac2ax=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}
Where a,ba,b and cc are the co-efficient.

Complete step by step solution:
In the above equation we have to find out the value of x.x.
So, first rearrange the equation as follow,
9x2324=0...(i)9{{x}^{2}}-324=0...(i)
The quadratic equation is as 9x2+bx+c=09{{x}^{2}}+bx+c=0 Where a,ba,b and cc are co-coefficient of the equation.
Now, equation with equation (i)(i)we get.
a=9,a=9, here the value of aa is 9.9.
b=0b=0, here the value of bb is 0.0.
c=324,c=-324, here the value of cc is 324-324
Now, substituting the value in quadratic form as,
x=b±b24ac2ax=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}
We get,
x=0±024×9×(324)2×9x=\dfrac{0\pm \sqrt{{{0}^{2}}-4\times 9\times \left( -324 \right)}}{2\times 9}
Further solving we get,
x=±(4×9×(324))18x=\dfrac{\pm \sqrt{-\left( 4\times 9\times \left( -324 \right) \right)}}{18}
Now multiplying 44 with 99 and 324-324, we have x=±1166418x=\dfrac{\pm \sqrt{11664}}{18}
Now dominating root, we get.
x=±10818x=\dfrac{\pm 108}{18}
Here we have two value for xx we get,
x=+10818x=\dfrac{+108}{18} and x=10818x=\dfrac{-108}{18}

Here we have two value for xx we get,x=+6x=+6 and x=6x=-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=2,2x+1=2, if we move 11 from left side to right side i.e. after equal to then the positive sign of +1+1 gets converted into negative sign.
Thus, it will equal to 2x=212x=2-1
Similarly, in 23x=4,2-3x=-4, If here we more 3x-3x from left side to right side then it will become positive, and if we move 4-4 from right side to left side it will become +4.+4.
So, 23x=43x=2+42-3x=-4\Rightarrow 3x=2+4
Similarly, if 2x=42x=4 then here 22 is in multiplication with x,x, in order to determine the value of xx we have to replace 22 from left side to right side, so it will become divided.
i.e. 2x=4x=422x=4\Rightarrow x=\dfrac{4}{2} here, 22 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,x=5, here 22 is in division with xx on the left side, so when we solve the equation then it will be transferred to the right side, and converted into multiplication.
Like, 12x=5x(5×2)\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 (2x,12x,3x)\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×3=112x\times 3=11
Here, as we have to determine the value of 2x,2x, As 33 is in addition with 2x2x in left side,
So, in order to neutralise it. Will subtract 33 from side,
So, equation becomes,
2x+33=1132x+3-3=11-3
As, 2x=8x=82=42x=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.