-3/2 x² -3/4x-1=0 using quadratic formula

To solve the quadratic equation \(-\frac{3}{2}x^2 - \frac{3}{4}x - 1 = 0\) using the quadratic formula, we first identify the coefficients \(a\), \(b\), and \(c\) from the general form of a quadratic equation \(ax^2 + bx + c = 0\).

Given:
\[
a = -\frac{3}{2}, \quad b = -\frac{3}{4}, \quad c = -1
\]

The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now, plug in the values:

1. **Calculate the discriminant** \(\Delta\):
   \[
   \Delta = b^2 - 4ac = \left(-\frac{3}{4}\right)^2 - 4\left(-\frac{3}{2}\right)\left(-1\right)
   \]
   \[
   \Delta = \frac{9}{16} - 4 \times \left(\frac{3}{2}\right) \times 1
   \]
   \[
   \Delta = \frac{9}{16} - 6 = \frac{9}{16} - \frac{96}{16} = \frac{-87}{16}
   \]

   Since the discriminant is negative, the solutions will be complex.

2. **Apply the quadratic formula**:
   \[
   x = \frac{-(-\frac{3}{4}) \pm \sqrt{\frac{-87}{16}}}{2 \times -\frac{3}{2}}
   \]
   Simplify the expression:
   \[
   x = \frac{\frac{3}{4} \pm \sqrt{\frac{-87}{16}}}{-3}
   \]
   Simplify further:
   \[
   x = \frac{\frac{3}{4} \pm \frac{\sqrt{-87}}{4}}{-3} = \frac{3 \pm \sqrt{-87}}{-12}
   \]

3. **Express the roots in the form of a complex number**:
   \[
   x = \frac{3}{-12} \pm \frac{\sqrt{87}i}{12}
   \]
   \[
   x = -\frac{1}{4} \pm \frac{\sqrt{87}}{12}i
   \]

So, the solutions are:
\[
x_1 = -\frac{1}{4} + \frac{\sqrt{87}}{12}i, \quad x_2 = -\frac{1}{4} - \frac{\sqrt{87}}{12}i
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How does the discriminant help determine the nature of the roots in a quadratic equation?
2. What is the significance of complex roots in quadratic equations?
3. How can we simplify the quadratic formula for specific types of quadratic equations?
4. What are the real-world applications of complex roots in quadratic equations?
5. How does the quadratic formula relate to completing the square?

**Tip:** Always check the discriminant before proceeding with the quadratic formula; it helps determine whether the roots are real, repeated, or complex.

Learn how to solve the quadratic equation -3/2 x² -3/4x-1 = 0 step-by-step with the quadratic formula. This problem involves complex numbers and requires careful calculation of the discriminant. Suitable for high school students studying quadratic equations.

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