https://file.aiquickdraw.com/mathbot/file/1728275606652ulf4h83i.jpg

The equation provided is:

\[
4z^2 + 3z + 10 = 6
\]

To solve this equation using the quadratic formula, we first need to rewrite it in standard form:

\[
4z^2 + 3z + 10 - 6 = 0
\]

This simplifies to:

\[
4z^2 + 3z + 4 = 0
\]

The quadratic formula is:

\[
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 4z^2 + 3z + 4 = 0 \), the coefficients are:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 4 \)

Now, plug these values into the quadratic formula:

1. Calculate the discriminant \( \Delta \):

\[
\Delta = b^2 - 4ac = 3^2 - 4(4)(4) = 9 - 64 = -55
\]

Since the discriminant is negative (\( \Delta = -55 \)), the solutions will be complex.

2. Solve for \( z \):

\[
z = \frac{-3 \pm \sqrt{-55}}{2(4)} = \frac{-3 \pm i\sqrt{55}}{8}
\]

So, the two solutions are:

\[
z = \frac{-3 + i\sqrt{55}}{8}, \quad z = \frac{-3 - i\sqrt{55}}{8}
\]

These are the solutions in terms of complex numbers.

Would you like more details or have any questions?

Here are some related questions:
1. How does the discriminant determine the nature of the roots?
2. What does it mean when the discriminant is negative?
3. Can we use other methods to solve quadratic equations, like completing the square?
4. What is the significance of complex numbers in quadratic equations?
5. Can quadratic equations always have real solutions?

**Tip**: When the discriminant is negative, the solutions are complex, involving imaginary numbers.

Solve equation by the quadratic formula. List the solutions, separated by commas.
4z^2 + 3z + 10 = 6

Learn how to solve the quadratic equation 4z^2 + 3z + 4 using the quadratic formula. This problem involves the discriminant and complex solutions, perfect for students in Grades 10-12.

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