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What are the solutions of this quadratic equation?

[tex]\[ 25x^2 + 1 = 0 \][/tex]

A. [tex]\( x = \pm \frac{1}{5}i \)[/tex]

B. [tex]\( x = 5 \pm i \)[/tex]

C. [tex]\( x = \pm 5i \)[/tex]

D. [tex]\( x = 1 \pm 5i \)[/tex]


Sagot :

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]