The image contains a list of quadratic equations and asks for the set of values of $k$ that lead to specific conditions on the roots (e.g., distinct real roots, equal roots, real roots, no real roots). The general approach to solving these problems involves applying the discriminant condition for quadratic equations.

Recall:

For a quadratic equation of the form $ax^2 + bx + c = 0$, the discriminant $\Delta$ is given by: $\Delta = b^2 - 4ac$

• Two distinct real roots: $\Delta > 0$
• Equal roots: $\Delta = 0$
• No real roots: $\Delta < 0$

Let’s go through each part based on the conditions given in the image:

i) $2x^2 - x + 3k = 0$ has two distinct real roots.

The equation is of the form $ax^2 + bx + c = 0$ where:

• $a = 2$
• $b = -1$
• $c = 3k$

The discriminant is: $\Delta = (-1)^2 - 4(2)(3k) = 1 - 24k$ For two distinct real roots, $\Delta > 0$: $1 - 24k > 0 \quad \Rightarrow \quad k < \frac{1}{24}$

Thus, $k < \frac{1}{24}$.

ii) $3x^2 + 5x - k = 0$ has two distinct real roots.

For this equation:

• $a = 3$
• $b = 5$
• $c = -k$

The discriminant is: $\Delta = 5^2 - 4(3)(-k) = 25 + 12k$ For two distinct real roots, $\Delta > 0$: $25 + 12k > 0 \quad \Rightarrow \quad k > -\frac{25}{12}$

Thus, $k > -\frac{25}{12}$.

iii) $5x^2 - 2x + (2k - 1) = 0$ has equal roots.

For this equation:

• $a = 5$
• $b = -2$
• $c = 2k - 1$

The discriminant is: $\Delta = (-2)^2 - 4(5)(2k - 1) = 4 - 20(2k - 1) = 4 - 40k + 20$ $\Delta = 24 - 40k$ For equal roots, $\Delta = 0$: $24 - 40k = 0 \quad \Rightarrow \quad k = \frac{24}{40} = \frac{3}{5}$

Thus, $k = \frac{3}{5}$.

iv) $2x^2 + 3x - (3k + 1) = 0$ has equal roots.

For this equation:

• $a = 2$
• $b = 3$
• $c = -(3k + 1)$

The discriminant is: $\Delta = 3^2 - 4(2)(-(3k + 1)) = 9 + 8(3k + 1) = 9 + 24k + 8 = 17 + 24k$ For equal roots, $\Delta = 0$: $17 + 24k = 0 \quad \Rightarrow \quad k = -\frac{17}{24}$

Thus, $k = -\frac{17}{24}$.

v) $-x^2 + 3x + (k + 1) = 0$ has real roots.

For this equation:

• $a = -1$
• $b = 3$
• $c = k + 1$

The discriminant is: $\Delta = 3^2 - 4(-1)(k + 1) = 9 + 4(k + 1) = 9 + 4k + 4 = 13 + 4k$ For real roots, $\Delta \geq 0$: $13 + 4k \geq 0 \quad \Rightarrow \quad k \geq -\frac{13}{4}$

Thus, $k \geq -\frac{13}{4}$.

Would you like more details on any of these solutions or additional explanations?

Follow-up questions:

1. How do you determine whether the roots of a quadratic are real, equal, or complex?
2. What are some practical applications of solving quadratic equations?
3. Can you explain how to solve a quadratic equation using the quadratic formula?
4. How does the sign of the discriminant affect the nature of the roots?
5. How can you solve quadratic inequalities using the discriminant?

Tip:

Always check the discriminant first before solving quadratic equations. It can give you a quick idea of the type of solutions the equation will have.