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What are the solutions of this quadratic equation?

[tex]\[ x^2 + 10 = 0 \][/tex]

A. [tex]\( x = \pm \sqrt{10} \)[/tex]

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

C. [tex]\( x = \pm \sqrt{10} i \)[/tex]

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


Sagot :

To solve the quadratic equation [tex]\( x^2 + 10 = 0 \)[/tex], let's follow the steps:

1. Isolate the quadratic term: We need to isolate [tex]\( x^2 \)[/tex] on one side of the equation. We can do this by subtracting 10 from both sides:
[tex]\[ x^2 + 10 - 10 = 0 - 10 \][/tex]
Simplifying this, we get:
[tex]\[ x^2 = -10 \][/tex]

2. Take the square root of both sides: To solve for [tex]\( x \)[/tex], we take the square root of both sides of the equation. Remember, when taking the square root of a negative number, we use the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex]:
[tex]\[ x = \pm \sqrt{-10} \][/tex]

3. Simplify the square root: The square root of [tex]\(-10\)[/tex] can be written using the imaginary unit [tex]\( i \)[/tex]:
[tex]\[ x = \pm \sqrt{10} \cdot \sqrt{-1} = \pm \sqrt{10} i \][/tex]

Therefore, the solutions to the equation [tex]\( x^2 + 10 = 0 \)[/tex] are:
[tex]\[ x = \pm \sqrt{10} i \][/tex]

By examining the given answer choices, the correct one is:

C. [tex]\( x = \pm \sqrt{10} i \)[/tex]