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How do you solve the equation $2{x^2} + 5 = 5{x^2} - 37?$

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: First of all we will take all the terms on one side of the equation and then find the pair of like terms among it and then will simplify the like terms and then will solve the simplified equation to get the value of the unknown term “x”.

Complete step-by-step solution:
Take the given expression: $2{x^2} + 5 = 5{x^2} - 37$
The above equation can be written as: $5{x^2} - 37 = 2{x^2} + 5$
Move all the terms on the left hand side of the equation from the right hand side of the equation. When you move any term from one side to another then the sign of the term also changes. Positive term becomes negative and negative term becomes positive.
$5{x^2} - 37 - 2{x^2} - 5 = 0$
Arrange the like terms together in the above equation.
$\underline {5{x^2} - 2{x^2}} - \underline {37 - 5} = 0$
Simplify among the like terms. When you simplify between both the terms having negative sign then you have to do addition and give sign of bigger number to the resultant value.
$3{x^2} - 42 = 0$
Move constant on the right hand side of the equation.
$3{x^2} = 42$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow {x^2} = \dfrac{{42}}{3}$
Find the factor on the numerator on the right hand side of the equation,
$ \Rightarrow {x^2} = \dfrac{{14 \times 3}}{3}$
Common factors from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.
$ \Rightarrow {x^2} = 14$
Take the square root on both the sides of the equation.
$ \Rightarrow \sqrt {{x^2}} = \sqrt {14} $
Square and square root cancel each other on the left hand side of the equation.
$ \Rightarrow x = \pm \sqrt {14} $
This is the required solution.

Note: Always remember that the square of positive or the negative number always gives us the positive number but the square root of positive number gives positive or the negative number therefore, we have kept plus or minus ahead of the number. Be careful about the plus or minus signs.

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