Mathematics, 29.05.2021 20:00, MrKrinkle77

Express the following in the form a +bi, where a and b are real numbers: $\sqrt{24 + 10i}$

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answered: isiahamccoy8822

5+i

Another accepted answer is -5-i, but if your teacher wants only one answer, then I'd go for 5+i

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Work Shown:

$\sqrt{24+10i} = a+bi\\\\\left(\sqrt{24+10i}\right)^2 = (a+bi)^2\\\\24+10i = a^2+2abi+b^2i^2\\\\24+10i = a^2+2abi+b^2(-1)\\\\24+10i = a^2+2abi-b^2\\\\24+10i = (a^2-b^2)+(2ab)i\\\\$

Equating terms, we have this system

$\begin{cases}24 = a^2-b^2\ \text{.... real terms}\\10 = 2ab\ \text{.... imaginary terms}\end{cases}$

Solve the second equation for b to get b = 5/a

Plug that into the first equation to solve for 'a'

$24 = a^2-b^2\\\\24 = a^2-\left(\frac{5}{a}\right)^2\\\\24 = a^2-\frac{25}{a^2}\\\\24a^2 = a^4-25\\\\0 = a^4-24a^2-25\\\\a^4-24a^2-25 = 0\\\\(a^2-25)(a^2+1) = 0\\\\(a-5)(a+5)(a^2+1) = 0\\\\$

Setting each factor equal to zero would lead to...

a-5 = 0 solves to a = 5a+5 = 0 solves to a = -5a^2+1 = 0 solves to a = i and a = -i

We're told that 'a' is a real number, so we ignore the solutions "a = i and a = -i". The only possible solutions are a = 5 and a = -5

If a = 5, then,

b = 5/a = 5/5 = 1

$\sqrt{24+10i} = a+bi = 5+1i = 5+i$

or in short,

$\sqrt{24+10i} = 5+i$

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If a = -5, then b = 5/a = 5/(-5) = -1

So it's very possible that we could also say

$\sqrt{24+10i} = -5-i\\\\$

If you wanted to combine the two we would use the plus/minus notation like so

$\sqrt{24+10i} = \pm(5+i)\\\\$

This is due to (5+i)^2 and (-5-i)^2 both having the same result of 24+10i. Hence the plus/minus. If your teacher wants one answer only, then I'd go for 5+i, as we could consider it a "principal" square root in a sense.

answered: Guest

so the possibilities of the measurement of the side of the triangle are either it is longest side or not the longest side

when it is the longest side

s < 25 + 35s < 60

if it is not the longest side

35 < 25 + s10 < sso 10 < s < 60so the perimeter70 < p < 130

answered: Guest

the degrees would remain the same ( the number to the left of the decimal point.

the multiply the decimal by 60 to get the minutes and use the numbers to the left of the decimal point.

then multiply the decimal portion of that by 60 for the seconds and round off as needed.

39 .956652

the 39 remains the degrees.