**Solve the equation x**^{2}+ 3 = 0

Solution:

The given quadratic equation is x^{2} + 3 = 0

On comparing the given equation with ax^{2} + bx + c = 0,

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

Discriminant D = b^{2} – 4ac = 0^{2} – 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 2x ^{2} + x + 1 = 0**

Solution:

The given quadratic equation is 2x^{ 2} + x + 1 = 0

On comparing the given equation with ax^{2} + bx + c = 0,

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

Discriminant D = b^{2} – 4ac = 1^{2} – 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}

**Solve the equation x**^{2}+ 3x + 9 = 0

Solution:

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

On comparing the given equation with ax^{2} + bx + c = 0,

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

Discriminant D = b^{2} – 4ac = 3^{2} – 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 ax^{2 }+ bx + c = 0,

we obtain a = –1, b = 1, and c = –2

Discriminant D = b^{2} – 4ac = 1^{2} – 4 × (–1) × (–2) = 1 – 8 = –7

Therefore, the required solutions are

^{-b}^{ ±}^{ √D}/_{2a} = ^{-1 }^{±}^{ √-7}/_{2x(-1) } = ^{-1 }^{±}^{ √7 i}/_{2}

**Solve the equation x**^{2}+ 3x + 5 = 0

Solution:

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

On comparing the given equation with ax^{2} + bx + c = 0,

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

Discriminant D = b^{2} – 4ac = 3^{2} – 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 ax^{2} + bx + c = 0, we obtain a = 1, b = –1, and c = 2

Discriminant D = b^{2} – 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}

**Solve the equation √2x**^{2}+ x+ √2 = 0

Solution:

The given quadratic equation is √2x^{2} + x+ √2 = 0

On comparing the given equation with ax^{2} + bx + c = 0,

we get a =√2 , b = 1, and c = √2

Discriminant D = b^{2} – 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 √3x ^{2} – √2x + 3√3 = 0**

Solution:

The given quadratic equation is √3x^{2} – √2x + 3√3 = 0

On comparing the given equation with ax^{2} + bx + c = 0,

we have 𝑎 = √3, 𝑏 = −√2 𝑎𝑛𝑑 𝑐 = 3√3

Discriminant D = b^{2} – 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}

**Solve the equation x**^{2 }+ x +^{1}/_{√2}= 0

Solution:

The given quadratic equation x^{2 }+ x + ^{1}/_{√2} = 0

x^{2 }+ x + ^{1}/_{√2} = √2x^{2 }+ √2x + 1 = 0

On comparing this equation with ax^{2} + bx + c = 0, we get 𝑎 = √2, 𝑏 = √2 𝑎𝑛𝑑 𝑐 = 1

Discriminant (D) = b^{2} – 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}

**Solve the equation x**^{2}+^{x}/_{√2}+ 1 = 0

Solution:

The given quadratic equation is x^{2} + ^{x}/_{√2} + 1 = 0

x^{2} + ^{x}/_{√2} + 1 = √2x^{2} + x + √2 = 0

On comparing this equation with ax^{2} + bx + c = 0, we obtain 𝑎 = √2, 𝑏 = 1 𝑎𝑛𝑑 𝑐 = √2

Discriminant D = b^{2} – 4ac = 1^{2} – 4√2x√2 = 1 – 8 = -7

Therefore, the required solutions are

^{-b}^{ ±}^{ √D}/_{2a} = ^{-1 }^{±}^{ √-7}/_{2x√2} = ^{-1 }^{±}^{ √7 i}/_{2√2}