# Complex Numbers and Quadratic Equations – Exercise 5.3 – Class XI

1. Solve the equation x2 + 3 = 0

Solution:

The given quadratic equation is x2 + 3 = 0

On comparing the given equation with ax2 + bx + c = 0,

we get a = 1, b = 0, and c = 3

Discriminant D = b2 – 4ac = 02 – 4 × 1 × 3 = –12

Therefore, the required solutions are

-b ± √D/2a = ± √-12/2×1  = ± √12 i/2 = ± 2√3i/2 = ±√3i

2: Solve the equation 2x2 + x + 1 = 0

Solution:

The given quadratic equation is 2x 2 + x + 1 = 0

On comparing the given equation with ax2 + bx + c = 0,

we get a = 2, b = 1, and c = 1

Discriminant D = b2 – 4ac = 12 – 4 × 2 × 1 = 1 – 8 = –7

Therefore, the required solutions are

-b ± √D/2a = 1± √-7/2×2  = -1 ± √7 i/2×2 = -1 ± √7i/4

1. Solve the equation x2 + 3x + 9 = 0

Solution:

The given quadratic equation is x 2 + 3x + 9 = 0

On comparing the given equation with ax2 + bx + c = 0,

we obtain a = 1, b = 3, and c = 9

Discriminant D = b2 – 4ac = 32 – 4 × 1 × 9 = 9 – 36 = –27

Therefore, the required solutions are

-b ± √D/2a = -3 ± √-27/2×1  = -3 ± 3√-3/2 = -3 ± 3√3i/2

4: Solve the equation –x 2 + x – 2 = 0

Solution:

The given quadratic equation is –x 2 + x – 2 = 0

On comparing the given equation with ax2 + bx + c = 0,

we obtain a = –1, b = 1, and c = –2

Discriminant D = b2 – 4ac = 12 – 4 × (–1) × (–2) = 1 – 8 = –7

Therefore, the required solutions are

-b ± √D/2a = -1 ± √-7/2x(-1)  = -1 ± √7 i/2

1. Solve the equation x2 + 3x + 5 = 0

Solution:

The given quadratic equation is x 2 + 3x + 5 = 0

On comparing the given equation with ax2 + bx + c = 0,

we get a = 1, b = 3, and c = 5

Discriminant D = b2 – 4ac = 32 – 4 × 1 × 5 =9 – 20 = –11

Therefore, the required solutions are

-b ± √D/2a = -3 ± √-11/2×1  = -3 ± √11 i/2

6: Solve the equation x 2 – x + 2 = 0

Solution:

The given quadratic equation is x 2 – x + 2 = 0

On comparing the given equation with ax2 + bx + c = 0, we obtain a = 1, b = –1, and c = 2

Discriminant D = b2 – 4ac = (–1)2 – 4 × 1 × 2 = 1 – 8 = –7

Therefore, the required solutions are

-b ± √D/2a = -(-1) ± √-7/2×1  = 1 ± √12 i/2

1. Solve the equation √2x2 + x+ √2 = 0

Solution:

The given quadratic equation is √2x2 + x+ √2 = 0

On comparing the given equation with ax2 + bx + c = 0,

we get a =√2 , b = 1, and c = √2

Discriminant D = b2 – 4ac = 12 – = 1 – 8 = –7

Therefore, the required solutions are

-b ± √D/2a = -1 ± √-7/2×1  = -1 ± √7 i/2

8: Solve the equation √3x2 – √2x + 3√3 = 0

Solution:

The given quadratic equation is √3x2 – √2x + 3√3 = 0

On comparing the given equation with ax2 + bx + c = 0,

we have 𝑎 = √3, 𝑏 = −√2 𝑎𝑛𝑑 𝑐 = 3√3

Discriminant D = b2 – 4ac = (-√2)2 – 4(√3)(3√3) = 2 – 36 = -34

Therefore, the required solutions are

-b ± √D/2a = -(-√2) ± √-34/2×1  = -(-√2) ± √34i/2

1. Solve the equation x2 + x + 1/√2 = 0

Solution:

The given quadratic equation x2 + x + 1/√2 = 0

x2 + x + 1/√2 = √2x2 + √2x + 1 = 0

On comparing this equation with ax2 + bx + c = 0, we get 𝑎 = √2, 𝑏 = √2 𝑎𝑛𝑑 𝑐 = 1

Discriminant (D) = b2 – 4ac = (√2)2 – 4 (√2)x1 = 2 – 4√2

Therefore, the required solutions are

-b ± √D/2a = -√2 ± √[2-4√2] /2x√2  = -√2 ± √[2(1-2√2] /2x√2 =  [[-√2 ±√2(√(2√2-1)]i/2x√2] = -1±(√2(√2-1))i/2

1. Solve the equation x2 + x/√2 + 1 = 0

Solution:

The given quadratic equation is x2 + x/√2 + 1 = 0

x2 + x/√2 + 1 = √2x2 + x + √2 = 0

On comparing this equation with ax2 + bx + c = 0, we obtain 𝑎 = √2, 𝑏 = 1 𝑎𝑛𝑑 𝑐 = √2

Discriminant D = b2 – 4ac = 12 – 4√2x√2 = 1 – 8 = -7

Therefore, the required solutions are

-b ± √D/2a = -1 ± √-7/2x√2 = -1 ± √7 i/2√2