Discover new perspectives and gain insights with IDNLearn.com's diverse answers. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.
Sagot :
To solve for the missing number in the equation where [tex]\( q = -i \sqrt{2} \)[/tex]:
[tex]\[ q^2 + \square = 0 \][/tex]
let's first square [tex]\( q \)[/tex]:
[tex]\[ q = -i \sqrt{2} \][/tex]
Squaring both sides, we get:
[tex]\[ q^2 = (-i \sqrt{2})^2 \][/tex]
We know that:
[tex]\[ (-i \sqrt{2})^2 = (i^2)(\sqrt{2})^2 = (-1)(2) = -2 \][/tex]
So:
[tex]\[ q^2 = -2 \][/tex]
To make the equation [tex]\( q^2 + \square = 0 \)[/tex] true, we need:
[tex]\[ q^2 + 2 = 0 \][/tex]
Thus, the missing number is [tex]\( 2 \)[/tex]. To verify, substitute:
[tex]\[ q^2 + 2 = 0 \implies -2 + 2 = 0 \][/tex]
Next, we need to find the two solutions of the equation:
[tex]\[ q^2 + 2 = 0 \][/tex]
First, rearrange it to standard form:
[tex]\[ q^2 = -2 \][/tex]
Taking the square root of both sides:
[tex]\[ q = \pm \sqrt{-2} \][/tex]
Recall that the square root of a negative number involves the imaginary unit [tex]\( i \)[/tex]:
[tex]\[ \sqrt{-2} = \sqrt{2} \cdot i \][/tex]
Thus, the two solutions are:
[tex]\[ q = i \sqrt{2} \quad \text{and} \quad q = -i \sqrt{2} \][/tex]
In summary, the two solutions in simplified, rationalized form are:
[tex]\[ q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2} \][/tex]
[tex]\[ q^2 + \square = 0 \][/tex]
let's first square [tex]\( q \)[/tex]:
[tex]\[ q = -i \sqrt{2} \][/tex]
Squaring both sides, we get:
[tex]\[ q^2 = (-i \sqrt{2})^2 \][/tex]
We know that:
[tex]\[ (-i \sqrt{2})^2 = (i^2)(\sqrt{2})^2 = (-1)(2) = -2 \][/tex]
So:
[tex]\[ q^2 = -2 \][/tex]
To make the equation [tex]\( q^2 + \square = 0 \)[/tex] true, we need:
[tex]\[ q^2 + 2 = 0 \][/tex]
Thus, the missing number is [tex]\( 2 \)[/tex]. To verify, substitute:
[tex]\[ q^2 + 2 = 0 \implies -2 + 2 = 0 \][/tex]
Next, we need to find the two solutions of the equation:
[tex]\[ q^2 + 2 = 0 \][/tex]
First, rearrange it to standard form:
[tex]\[ q^2 = -2 \][/tex]
Taking the square root of both sides:
[tex]\[ q = \pm \sqrt{-2} \][/tex]
Recall that the square root of a negative number involves the imaginary unit [tex]\( i \)[/tex]:
[tex]\[ \sqrt{-2} = \sqrt{2} \cdot i \][/tex]
Thus, the two solutions are:
[tex]\[ q = i \sqrt{2} \quad \text{and} \quad q = -i \sqrt{2} \][/tex]
In summary, the two solutions in simplified, rationalized form are:
[tex]\[ q = -i \sqrt{2} \quad \text{and} \quad q = i \sqrt{2} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.