solve the quadratic: x to the second power + 2x - 4 = -9 what is the discriminant? the quadratic has (2 irrational, 2 rational, 1 rational, or 2 complex) solutions the smaller solution is _______ the bigger solution is ________

We start by solving the quadratic equation \( x^2 + 2x - 4 = -9 \). First, let's rearrange it to the standard quadratic form:

\[
x^2 + 2x - 4 + 9 = 0
\]
\[
x^2 + 2x + 5 = 0
\]

This is now in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 5 \)

### Step 1: Compute the discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

Substituting the values of \( a \), \( b \), and \( c \):

\[
\Delta = (2)^2 - 4(1)(5)
\]
\[
\Delta = 4 - 20 = -16
\]

### Step 2: Nature of the solutions

The discriminant is \( -16 \), which is negative. When the discriminant is negative, the quadratic equation has two **complex** (non-real) solutions.

### Step 3: Find the solutions using the quadratic formula

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\]

Substitute \( b = 2 \), \( a = 1 \), and \( \Delta = -16 \):

\[
x = \frac{-2 \pm \sqrt{-16}}{2(1)}
\]
\[
x = \frac{-2 \pm 4i}{2}
\]
\[
x = \frac{-2}{2} \pm \frac{4i}{2}
\]
\[
x = -1 \pm 2i
\]

### Step 4: Conclusion

- The discriminant is \( -16 \).
- The quadratic equation has **2 complex** solutions: \( -1 + 2i \) and \( -1 - 2i \).
- The smaller solution is \( -1 - 2i \).
- The bigger solution is \( -1 + 2i \).

Do you want more details or have any questions?

Here are 5 related questions:
1. What happens when the discriminant is zero?
2. How does the discriminant affect the number of real solutions?
3. Can you explain why complex solutions come in conjugate pairs?
4. How do you graph quadratic functions with complex solutions?
5. What role does the quadratic formula play in finding irrational solutions?

**Tip:** The discriminant not only tells us the number of solutions but also whether those solutions are real or complex!

Learn how to solve the quadratic equation x^2 + 2x + 5 = 0, find the discriminant, and determine the nature of the solutions. This step-by-step guide covers algebraic concepts like the quadratic formula and the discriminant theorem, aimed at students in Grades 9-12.

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