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.