(use the value’s a a,b and c in finding the roots of the quadratic equations that have two x in the sum and the product of roots of quadratic equations) x² + 7x - 18 =

To find the roots of the quadratic equation \( x^2 + 7x - 18 = 0 \) using the values of \( a \), \( b \), and \( c \), we follow the general form of a quadratic equation:

\[
ax^2 + bx + c = 0
\]

Here:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 7 \) (coefficient of \( x \))
- \( c = -18 \) (constant term)

### Step 1: Find the sum and product of the roots
From the quadratic formula, the sum and product of the roots can be determined directly from \( a \), \( b \), and \( c \):

1. **Sum of the roots** \( (\alpha + \beta) \) is given by:
   \[
   \alpha + \beta = -\frac{b}{a} = -\frac{7}{1} = -7
   \]

2. **Product of the roots** \( (\alpha \cdot \beta) \) is given by:
   \[
   \alpha \cdot \beta = \frac{c}{a} = \frac{-18}{1} = -18
   \]

### Step 2: Find the roots using the quadratic formula
The roots of the quadratic equation are given by the quadratic formula:

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

Substituting the values of \( a = 1 \), \( b = 7 \), and \( c = -18 \):

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(-18)}}{2(1)}
\]

\[
x = \frac{-7 \pm \sqrt{49 + 72}}{2}
\]

\[
x = \frac{-7 \pm \sqrt{121}}{2}
\]

\[
x = \frac{-7 \pm 11}{2}
\]

Now, calculate the two possible roots:

1. For \( x_1 \):
   \[
   x_1 = \frac{-7 + 11}{2} = \frac{4}{2} = 2
   \]

2. For \( x_2 \):
   \[
   x_2 = \frac{-7 - 11}{2} = \frac{-18}{2} = -9
   \]

### Final Roots:
- \( x_1 = 2 \)
- \( x_2 = -9 \)

### Summary:
The roots of the quadratic equation \( x^2 + 7x - 18 = 0 \) are \( 2 \) and \( -9 \).

---

Would you like further details on any part of the process or explanations? 

Here are some related questions to explore:

1. How does the quadratic formula work in different cases?
2. What happens when the discriminant is negative?
3. Can we always use the sum and product of roots to check the answer?
4. How can you complete the square to solve quadratic equations?
5. What are the applications of quadratic equations in real-life situations?

**Tip**: Always check the discriminant (\( b^2 - 4ac \)) to determine the nature of the roots (real or complex).

Use the values a, b, and c in finding the roots of the quadratic equation x² + 7x - 18 = 0.

Learn how to solve the quadratic equation x² + 7x - 18 using the quadratic formula. This solution involves key algebraic concepts such as finding the sum and product of the roots. Suitable for students in Grades 9-10.

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