To solve the quadratic equation:

\[
7x^2 - 19x + 10 = 0
\]

we'll use the quadratic formula:

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

In this equation:
- \( a = 7 \)
- \( b = -19 \)
- \( c = 10 \)

### Step 1: Calculate the Discriminant
The discriminant is given by \( b^2 - 4ac \):

\[
b^2 - 4ac = (-19)^2 - 4 \cdot 7 \cdot 10
\]
\[
= 361 - 280
\]
\[
= 81
\]

### Step 2: Apply the Quadratic Formula
Now we substitute into the quadratic formula:

\[
x = \frac{-(-19) \pm \sqrt{81}}{2 \cdot 7}
\]
\[
= \frac{19 \pm 9}{14}
\]

### Step 3: Solve for the Two Values of \( x \)

1. **For \( x = \frac{19 + 9}{14} \):**

   \[
   x = \frac{28}{14} = 2
   \]

2. **For \( x = \frac{19 - 9}{14} \):**

   \[
   x = \frac{10}{14} = \frac{5}{7}
   \]

### Solution
The solutions to the equation are:

\[
x = 2 \quad \text{and} \quad x = \frac{5}{7}
\]

Would you like further details or have any questions?

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**Related Questions:**
1. How can we determine the nature of the roots of a quadratic equation?
2. What are some alternative methods for solving quadratic equations?
3. How do we factor a quadratic equation if it’s factorable?
4. What does it mean if the discriminant is negative?
5. How would the solutions change if \( a \), \( b \), or \( c \) were different?

**Tip:** The discriminant \( b^2 - 4ac \) tells you whether the solutions are real or complex. If it's positive, the equation has two real solutions.

Learn how to solve the quadratic equation 7x^2 - 19x + 10 = 0 with a detailed, step-by-step guide. This solution covers important algebraic concepts such as the quadratic formula and discriminant, providing clear explanations for students in Grades 9-10.

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