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Let's go through the questions one by one and solve them.

### Question 6:
The quadratic equation is given as \(x^2 - 4x + 5 = 0\). You are asked to find \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\), where \(\alpha\) and \(\beta\) are the roots.

Using the sum and product of roots formulas:
- Sum of roots: \(\alpha + \beta = 4\)
- Product of roots: \(\alpha \beta = 5\)

We know that:
\[
\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta}
\]
Now, \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta = 4^2 - 2(5) = 16 - 10 = 6\).

Therefore:
\[
\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{6}{5}
\]

The correct answer is **d) \(\frac{6}{5}\)**.

---

### Question 7:
The quadratic equation is \(2x^2 + 3x - 1 = 0\). To solve for the roots, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = 3\), and \(c = -1\). Substituting these values into the formula:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-1)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 8}}{4} = \frac{-3 \pm \sqrt{17}}{4}
\]

The correct answer is **d) \(x = \frac{-3 \pm \sqrt{17}}{4}\)**.

---

### Question 8:
The quadratic equation is \(x^2 + 3x - 5 = 0\), and we are asked to find \(x_1^2 + x_2^2\), where \(x_1\) and \(x_2\) are the roots.

Using the identity:
\[
x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1x_2
\]
The sum of the roots \(x_1 + x_2 = -3\) and the product of the roots \(x_1x_2 = -5\).

Substituting these values:
\[
x_1^2 + x_2^2 = (-3)^2 - 2(-5) = 9 + 10 = 19
\]

The correct answer is **b) 19**.

---

### Question 9:
The equation is \((x - 3)^2 - 2(x - 3) = 0\). Let's make a substitution: let \(y = x - 3\). This transforms the equation into:
\[
y^2 - 2y = 0 \quad \Rightarrow \quad y(y - 2) = 0
\]
So, \(y = 0\) or \(y = 2\). Substituting back \(x - 3 = y\):
- If \(y = 0\), then \(x = 3\)
- If \(y = 2\), then \(x = 5\)

The correct answer is **b) 3 and 5**.

---

### Question 10:
The quadratic equation is \(x^2 + 3x - 10 = 0\). We need to analyze the discriminant, \(\Delta = b^2 - 4ac\).

For the equation \(x^2 + 3x - 10 = 0\), we have \(a = 1\), \(b = 3\), and \(c = -10\). The discriminant is:
\[
\Delta = 3^2 - 4(1)(-10) = 9 + 40 = 49
\]
Since \(\Delta > 0\), the roots are real and distinct. The quadratic has two real roots, and one of the roots can be found using the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{49}}{2} = \frac{-3 \pm 7}{2}
\]
So, \(x = 2\) or \(x = -5\).

The correct answer is **d) Salah satu akar PK adalah -5**.

---

Do you need more details or have any questions? 

Here are five related questions:
1. Can you explain the sum and product of roots for a quadratic equation in general form?
2. How is the discriminant used to determine the nature of the roots?
3. Why is the quadratic formula effective for solving any quadratic equation?
4. Can we always factor a quadratic equation instead of using the quadratic formula?
5. How can completing the square provide an alternative method to solve quadratics?

**Tip:** For quick verification of the nature of roots, always check the discriminant value before solving the full quadratic equation.

The image contains several quadratic equation problems, including solving quadratic equations using the quadratic formula, finding the sum and product of roots, and analyzing the discriminant.

Solve quadratic equations using the quadratic formula, find the sum and product of roots, and analyze the discriminant with detailed step-by-step solutions. Ideal for students in grades 9-12.

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