To find the complex number that has an absolute value of 5, let's analyze each of the provided options. The absolute value (or modulus) of a complex number [tex]\( a + bi \)[/tex] is given by the formula [tex]\( |a + bi| = \sqrt{a^2 + b^2} \)[/tex]. We will calculate the absolute value for each given complex number.
1. For [tex]\(-3 + 4i\)[/tex]:
[tex]\[
| -3 + 4i | = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
2. For [tex]\(2 + 3i\)[/tex]:
[tex]\[
| 2 + 3i | = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605
\][/tex]
3. For [tex]\(7 - 2i\)[/tex]:
[tex]\[
| 7 - 2i | = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.280
\][/tex]
4. For [tex]\(9 + 4i\)[/tex]:
[tex]\[
| 9 + 4i | = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.849
\][/tex]
From the calculations above, we see that the complex number [tex]\(-3 + 4i\)[/tex] has an absolute value of 5. Therefore, the complex number with an absolute value of 5 is:
[tex]\[
\boxed{-3+4i}
\][/tex]
