x^2+5x+5=0 solve this equation using this formula D=b^2-4ac

To solve the quadratic equation \( x^2 + 5x + 5 = 0 \) using the discriminant formula \( D = b^2 - 4ac \), let's follow these steps:

### 1. Identify the coefficients:
For the equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 5 \) (coefficient of \( x \))
- \( c = 5 \) (constant term)

### 2. Calculate the discriminant \( D \):
\[
D = b^2 - 4ac
\]
Substitute the values of \( a \), \( b \), and \( c \):
\[
D = 5^2 - 4(1)(5) = 25 - 20 = 5
\]

### 3. Interpret the discriminant:
Since \( D > 0 \), the equation has two distinct real roots.

### 4. Solve for the roots using the quadratic formula:
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
Substitute the values of \( b \), \( D \), and \( a \):
\[
x = \frac{-5 \pm \sqrt{5}}{2(1)} = \frac{-5 \pm \sqrt{5}}{2}
\]

### 5. Final Answer:
The solutions are:
\[
x_1 = \frac{-5 + \sqrt{5}}{2}, \quad x_2 = \frac{-5 - \sqrt{5}}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. How would the solutions change if the discriminant \( D \) were negative?
2. Can you calculate the exact decimal values of the roots?
3. How would you find the vertex of the parabola represented by this quadratic equation?
4. What is the sum and product of the roots of the equation?
5. How does the discriminant determine the nature of the roots?

**Tip:** The discriminant \( D \) not only helps in finding the roots but also tells you about the nature of those roots—whether they are real and distinct, real and equal, or complex.

Learn how to solve the quadratic equation x^2 + 5x + 5 = 0 using the discriminant formula D = b^2 - 4ac. This problem involves calculating the discriminant, identifying the nature of the roots, and applying the quadratic formula. Suitable for students in Grades 10-12.

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