Sure, let's solve the quadratic equation step by step.
The given quadratic equation is:
[tex]\[ 25x^2 + 1 = 0 \][/tex]
Step 1: Move the constant term to the other side of the equation.
[tex]\[ 25x^2 = -1 \][/tex]
Step 2: Divide both sides of the equation by 25 to isolate [tex]\( x^2 \)[/tex].
[tex]\[ x^2 = -\frac{1}{25} \][/tex]
Step 3: Take the square root of both sides of the equation to solve for [tex]\( x \)[/tex]. Remember that taking the square root of a negative number introduces imaginary numbers, denoted as [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].
[tex]\[ x = \pm \sqrt{-\frac{1}{25}} \][/tex]
Step 4: Simplify the square root of the fraction.
[tex]\[ x = \pm \frac{\sqrt{-1}}{\sqrt{25}} \][/tex]
Step 5: Recognize that [tex]\( \sqrt{-1} = i \)[/tex] and [tex]\( \sqrt{25} = 5 \)[/tex].
[tex]\[ x = \pm \frac{1}{5} i \][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[ x = \pm \frac{1}{5} i \][/tex]
From the given options, the correct answer is:
[tex]\[ x = \pm \frac{1}{5} i \][/tex]
