To determine which of the given complex numbers has an absolute value of 5, we can calculate the absolute value (or magnitude) of each complex number. The absolute value of a complex number [tex]\(a + bi\)[/tex] is given by the formula:
[tex]\[
|a + bi| = \sqrt{a^2 + b^2}
\][/tex]
Let's calculate the absolute value for each complex number provided:
1. For the complex number [tex]\(-3 + 4i\)[/tex]:
[tex]\[
|-3 + 4i| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
2. For the complex number [tex]\(2 + 3i\)[/tex]:
[tex]\[
|2 + 3i| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989
\][/tex]
3. For the complex number [tex]\(7 - 2i\)[/tex]:
[tex]\[
|7 - 2i| = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.280109889280518
\][/tex]
4. For the complex number [tex]\(9 + 4i\)[/tex]:
[tex]\[
|9 + 4i| = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.848857801796104
\][/tex]
Among the calculated absolute values, the number [tex]\(-3 + 4i\)[/tex] has an absolute value of 5. Thus, the complex number which has an absolute value of 5 is:
[tex]\[
\boxed{-3 + 4i}
\][/tex]
