Questions: Solve for (x). [ x^2-51=-54 ] Write your answer in simplified, rationalized form. [ x= text or x= ]

Transcript text: Solve for $x$. \[ x^{2}-51=-54 \] Write your answer in simplified, rationalized form. \[ x=\square \text { or } x=\square \]

Solution

Step 1: Set the Equation to Zero

Start by moving all terms to one side of the equation to set it to zero:

\[ x^2 - 51 + 54 = 0 \]

This simplifies to:

\[ x^2 + 3 = 0 \]

Step 2: Solve for $ x^2 $

Rearrange the equation to solve for $ x^2 $:

\[ x^2 = -3 \]

Step 3: Solve for $ x $

Take the square root of both sides to solve for $ x $. Remember that taking the square root of a negative number involves imaginary numbers:

\[ x = \pm \sqrt{-3} = \pm i\sqrt{3} \]

The solutions for $ x $ are:

\[ \boxed{x = i\sqrt{3} \text{ or } x = -i\sqrt{3}} \]

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