To solve the quadratic equation $5x^2 - x - k = 0$ given that the difference between its solutions is 1.8, we can use the properties of quadratic equations.

For a quadratic equation of the form $ax^2 + bx + c = 0$, the difference $d$ between the roots can be expressed as:

$d = \frac{\sqrt{b^2 - 4ac}}{a}$

In this case, $a = 5$, $b = -1$, and $c = -k$. Thus, the difference between the roots is:

$\frac{\sqrt{(-1)^2 - 4 \cdot 5 \cdot (-k)}}{5} = 1.8$

This simplifies to:

$\frac{\sqrt{1 + 20k}}{5} = 1.8$

Multiplying both sides by 5:

$\sqrt{1 + 20k} = 9$

Now, squaring both sides:

$1 + 20k = 81$

Solving for $k$:

$20k = 80 \quad \Rightarrow \quad k = 4$

Now, we can substitute $k$ back into the original equation:

$5x^2 - x - 4 = 0$

To find the roots, we apply the quadratic formula:

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

Substituting $a = 5$, $b = -1$, and $c = -4$:

$x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 5 \cdot (-4)}}{2 \cdot 5}$

Calculating the discriminant:

$x = \frac{1 \pm \sqrt{1 + 80}}{10} = \frac{1 \pm \sqrt{81}}{10} = \frac{1 \pm 9}{10}$

This gives us two solutions:

1. $x_1 = \frac{10}{10} = 1$
2. $x_2 = \frac{-8}{10} = -0.8$

Thus, the solutions are $x = 1$ and $x = -0.8$.

Summary of the Solutions:

• The solutions of the equation are $x_1 = 1$ and $x_2 = -0.8$.

Would you like further details or clarification? Here are some follow-up questions:

1. How do the roots relate to the graph of the quadratic function?
2. Can you explain the significance of the discriminant in finding the roots?
3. What happens to the roots if $k$ is increased further?
4. How would the roots change if the equation was $5x^2 + x - k = 0$ instead?
5. Could you derive the vertex of the parabola represented by the equation?

Tip: Always check the discriminant to understand the nature of the roots in a quadratic equation.