Question: User: Solution
Given the quadratic equation:
$$9z^2+16=0$$
Step 1: Rewrite the equation in standard form
Standard form for a quadratic equation is:
$$az^2+bz+c=0$$
Here, $$a=9$$, $$b=0$$, and $$c=16$$.
The equation is:
$$9z^2+0\cdot z+16=0$$
Step 2: Apply the quadratic formula
The quadratic formula is:
$$z=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$
Substituting the values of $$a$$, $$b$$, and $$c$$:
$$z=\frac{-0\pm \sqrt{0^2-4(9)(16)}}{2(9)}$$
$$z=\frac{\pm \sqrt{-576}}{18}$$
Step 3: Simplify the discriminant
Since the discriminant is negative, the solutions are complex:
$$-576=576\cdot (-1)=24^2 \cdot (-1)$$
$$\sqrt{-576}=24i$$
Step 4: Write the solution
Substitute back:
$$z=\frac{\pm 24i}{18}=\pm \frac{24}{18}i=\pm \frac{4}{3}i$$
Final answer
The solution set in simplest exact form is:
$$z=\pm \frac{4}{3}i$$
1. **State the problem:** Solve the quadratic equation $$9z^2 + 16 = 0$$.
2. **Rewrite in standard form:** The standard form is $$az^2 + bz + c = 0$$. Here, $$a=9$$, $$b=0$$, and $$c=16$$, so the equation is $$9z^2 + 0\cdot z + 16 = 0$$.
3. **Use the quadratic formula:** $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
4. **Substitute values:** $$z = \frac{-0 \pm \sqrt{0^2 - 4 \times 9 \times 16}}{2 \times 9} = \frac{\pm \sqrt{-576}}{18}$$.
5. **Simplify the discriminant:** Since $$-576 = 576 \times (-1) = 24^2 \times (-1)$$, we have $$\sqrt{-576} = 24i$$.
6. **Write the solutions:** $$z = \frac{\pm 24i}{18} = \pm \frac{24}{18}i = \pm \frac{4}{3}i$$.
7. **Final answer:** The solutions are $$z = \pm \frac{4}{3}i$$.
This means the quadratic has two complex conjugate solutions.