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How do you solve using the square root method \[{x^2} + 5 = 86\]?

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Last updated date: 22nd Jul 2024
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Answer
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Hint: Here, in the question, we have been given a quadratic equation and we are asked to solve for it using completing the square method. To solve the equation means that we have to find the value of the variable present in the equation. Completing the square method is one of the various methods of determining the roots of the quadratic equation.

Complete step-by-step solution:
Given equation \[{x^2} + 5 = 86\]
As the coefficient of \[{x^2}\] is already one, we don’t need to divide here with the coefficient. And the coefficient of \[x\] is also zero. So, we don’t need to divide the coefficient of \[x\] as well. Now, we will take the constant terms on one side of the equation i.e. the right hand side of the equation.
So, we have \[{x^2} + 5 = 86\]
Subtracting \[5\] both sides, we get,
\[{x^2} + 5 - 5 = 86 - 5\]
Rewriting the above equation,
\[{x^2} = 81\]
Taking square root both sides, we get,
\[\sqrt {{x^2}} = \sqrt {81} \]
Simplifying it, we get,
\[ x = \pm \sqrt {81} \\
   \Rightarrow x = \pm \sqrt {{{\left( 9 \right)}^2}} \\
   \Rightarrow x = \pm 9 \]
Therefore, \[x = 9\] and \[x = - 9\].
Thus, we got two values for \[x\] after solving it using the completing square method.
Hence we got \[x = 9\] and \[x = - 9\].

Note: In the equation given in the question here, there is no difference if we solve it using the completing square method or without using this method. Because the coefficient of \[x\] is zero here, therefore, square completion cannot be done here particularly the way we do in this method.
One thing to keep in mind when we face such types of questions, when finding square roots, is to take both the values positive and negative. Sometimes, we skip the negative value in a hurry.