## 1. The general form of a quadratic equation is:

### Answer:

### Explanation:

A quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

## 2. The roots of the equation x^2 – 6x + 9 = 0 are:

### Answer:

### Explanation:

The given equation is a perfect square trinomial. It can be written as (x-3)(x-3) = 0. Hence, the roots are 3 and 3.

## 3. If the quadratic equation px^2 + 6x + r = 0 has equal roots, then:

### Answer:

### Explanation:

For equal roots, discriminant b^2 – 4ac = 0. Plugging in the values, we get 36 – 4pr = 0.

## 4. The quadratic equation whose roots are 4 and -5 is:

### Answer:

### Explanation:

Using the sum and product of roots, the equation is x^2 – (sum of roots)x + product of roots = 0.

## 5. The nature of the roots of the equation 2x^2 – 8x + 8 = 0 is:

### Answer:

### Explanation:

Discriminant = b^2 – 4ac = 64 – 64 = 0. Hence, roots are real and equal.

## 6. For the quadratic equation ax^2 + bx + c = 0, if a > 0 and the discriminant is greater than 0, the parabola:

### Answer:

### Explanation:

Positive 'a' means the parabola opens upwards and discriminant > 0 means real and distinct roots.

## 7. The sum of the roots of the equation 3x^2 + 5x – 2 = 0 is:

### Answer:

### Explanation:

Sum of the roots = -b/a = -5/3.

## 8. If one root of the quadratic equation kx^2 – 14x + 8 = 0 is 2, then the value of k is:

### Answer:

### Explanation:

For x=2 to be a root, 2k – 14 + 8 = 0. Solving for k, we get k = 5.

## 9. The product of the roots of the equation x^2 + x + 1 = 0 is:

### Answer:

### Explanation:

Product of the roots = c/a = 1/1 = 1.

## 10. For the equation ax^2 + bx + c = 0, if one root is the square of the other, then:

### Answer:

### Explanation:

Let the roots be α and α^2. Sum = α + α^2 = -b/a and product = α^3 = c/a. Using these relations, we get c = b^2/4a.