To solve the equation [tex]\( 16x^2 + 9 = 0 \)[/tex], let's go through the steps one by one.
1. Set up the equation:
[tex]\[ 16x^2 + 9 = 0 \][/tex]
2. Isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ 16x^2 + 9 = 0 \][/tex]
Subtract 9 from both sides to get:
[tex]\[ 16x^2 = -9 \][/tex]
3. Solve for [tex]\( x^2 \)[/tex]:
Divide both sides by 16:
[tex]\[ x^2 = \frac{-9}{16} \][/tex]
4. Introduce the imaginary unit [tex]\( i \)[/tex]:
Since we have a negative number on the right side of the equation, this implies the solution will involve the imaginary unit [tex]\( i \)[/tex]. Recall that [tex]\( \sqrt{-1} = i \)[/tex].
5. Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{-9}{16}} \][/tex]
[tex]\[ x = \pm \frac{\sqrt{-9}}{\sqrt{16}} \][/tex]
6. Simplify the square root expression:
[tex]\[ \sqrt{-9} = \sqrt{-1 \cdot 9} = \sqrt{-1} \cdot \sqrt{9} = i \cdot 3 = 3i \][/tex]
[tex]\[ \sqrt{16} = 4 \][/tex]
Hence,
[tex]\[ x = \pm \frac{3i}{4} \][/tex]
Thus, the solutions to the equation [tex]\( 16x^2 + 9 = 0 \)[/tex] are:
[tex]\[ x = \pm \frac{3i}{4} \][/tex]
In numeric form, these roots are [tex]\( 0.75i \)[/tex] and [tex]\( -0.75i \)[/tex].
So, the answers are:
[tex]\[ x = \frac{3i}{4} \][/tex]
and
[tex]\[ x = -\frac{3i}{4} \][/tex]
