EXERCISE 10 (a) Discuss the nature of the roots of the following quadratic equations, without solving the equations: \( \begin{array}{llll}\text { (1) } x^{2}=0 & \text { (2) } x^{2}=16 & \text { (3) } x^{2}=7 \\ \text { (4) } x^{2}=-4 & \text { (5) }(x-2)^{2}=4 & \text { (6) }(x+3)^{2}=12 \\ \text { (7) } \quad(x+1)^{2}=0 & \text { (8) }(x-5)^{2}=-1 & \text { (9) } x^{2}-9=0\end{array} \)

To discuss the nature of the roots of the given quadratic equations, we can analyze each equation based on the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). The nature of the roots can be determined by the discriminant \( D = b^2 - 4ac \). 1. **Equation (1): \( x^2 = 0 \)** - This can be rewritten as \( x^2 - 0 = 0 \) (where \( a = 1, b = 0, c = 0 \)). - Discriminant: \( D = 0^2 - 4(1)(0) = 0 \). - **Nature of roots**: One real root (a double root). 2. **Equation (2): \( x^2 = 16 \)** - Rewritten as \( x^2 - 16 = 0 \) (where \( a = 1, b = 0, c = -16 \)). - Discriminant: \( D = 0^2 - 4(1)(-16) = 64 \). - **Nature of roots**: Two distinct real roots. 3. **Equation (3): \( x^2 = 7 \)** - Rewritten as \( x^2 - 7 = 0 \) (where \( a = 1, b = 0, c = -7 \)). - Discriminant: \( D = 0^2 - 4(1)(-7) = 28 \). - **Nature of roots**: Two distinct real roots. 4. **Equation (4): \( x^2 = -4 \)** - Rewritten as \( x^2 + 4 = 0 \) (where \( a = 1, b = 0, c = 4 \)). - Discriminant: \( D = 0^2 - 4(1)(4) = -16 \). - **Nature of roots**: Two complex roots (no real roots). 5. **Equation (5): \( (x-2)^2 = 4 \)** - Rewritten as \( (x-2)^2 - 4 = 0 \) (which expands to \( x^2 - 4x + 0 = 0 \)). - Discriminant: \( D = (-4)^2 - 4(1)(0) = 16 \). - **Nature of roots**: Two distinct real roots. 6. **Equation (6): \( (x+3)^2 = 12 \)** - Rewritten as \( (x+3)^2 - 12 = 0 \) (which expands to \( x^2 + 6x - 12 = 0 \)). - Discriminant: \( D = 6^2 - 4(1)(-12) = 36 + 48 = 84 \). - **Nature of roots**: Two distinct real roots. 7. **Equation (7): \( (x+1)^2 = 0 \)** - Rewritten as \( (x+1)^2 - 0 = 0 \) (which is already in standard form). - Discriminant: \( D = 0^2 - 4(1)(0) = 0 \). - **Nature of roots**: One real root (a double root). 8. **Equation (8): \( (x-5)^2 = -1 \)** - Rewritten as \( (x-5)^2 + 1 = 0 \) (which expands to \( x^2 - 10x + 26 = 0 \)). - Discriminant: \( D = (-10)^2 - 4(1)(26) = 100 - 104 = -4 \). - **Nature of roots**: Two complex roots (no real roots). 9. **Equation (9): \( x^2 - 9 = 0 \)** - This can be factored as \( (x-3)(x+3) = 0 \) (where \( a = 1, b = 0, c = -9 \)). - Discriminant: \( D = 0^2 - 4(1)(-9) = 36 \). - **Nature of roots**: Two distinct real roots. ### Summary of the Nature of Roots: 1. One real root (double root) 2. Two distinct real roots 3. Two distinct real roots 4. Two complex roots 5. Two distinct real roots 6. Two distinct real roots 7. One real root (double root) 8. Two complex roots 9. Two distinct real roots