x2+ 2x= - 5Find the Discriminant:

Nature of the roots:

Prove by solving using the quadratic formula:

We are given the quadratic equation:

\[
x^2 + 2x = -5
\]

### Step 1: Rewrite in standard form

To apply the quadratic formula, we need the equation in the form \( ax^2 + bx + c = 0 \).

\[
x^2 + 2x + 5 = 0
\]

Here, \(a = 1\), \(b = 2\), and \(c = 5\).

### Step 2: Find the discriminant

The discriminant (\(\Delta\)) is calculated using the formula:

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

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

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

### Step 3: Nature of the roots

The nature of the roots depends on the value of the discriminant:

- If \(\Delta > 0\), the roots are real and distinct.
- If \(\Delta = 0\), the roots are real and equal.
- If \(\Delta < 0\), the roots are complex and not real.

Since \(\Delta = -16\), which is less than 0, the equation has **complex** roots.

### Step 4: Solve using the quadratic formula

The quadratic formula is given by:

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

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

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

Simplify:

\[
x = \frac{-2 \pm 4i}{2}
\]

So, the two roots are:

\[
x = \frac{-2 + 4i}{2} = -1 + 2i
\]
\[
x = \frac{-2 - 4i}{2} = -1 - 2i
\]

### Conclusion:

- **Discriminant**: \(\Delta = -16\)
- **Nature of the roots**: Complex and not real
- **Roots**: \(x = -1 + 2i\) and \(x = -1 - 2i\)

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What happens when the discriminant is zero?
2. How can we graphically interpret complex roots in the complex plane?
3. How do we use the quadratic formula for equations with higher-degree polynomials?
4. Can complex roots ever appear as real numbers?
5. What are some practical applications of complex roots?

**Tip:** Always check the sign of the discriminant to determine whether the roots are real or complex before solving.

x^2 + 2x = -5. Find the Discriminant. Nature of the roots: Prove by solving using the quadratic formula.

Discover how to solve the quadratic equation x^2 + 2x + 5 = 0 using the discriminant and the quadratic formula. This detailed step-by-step solution explains how to find the discriminant (Δ = -16) and the complex roots of the equation. Suitable for students in Grades 9-11.

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