**Solve each of the following equations:**

**1. x ^{2} + 3 = 0**

x

^{2}+ 3 = 0

x

^{2}= – 3

x = ± √-3

This can be rewritten as following

x = ± √3 × √-1

Replacing √-1 with Imaginary Number iota (i)

⇒ x = ± √3i

Thus value of

**x**is

**± √3i**if its given that

**x**

^{2}+ 3 = 0**2. 2x ^{2} + x + 1 = 0**

Comparing

**2x**with standard quadratic equation

^{2}+ x + 1 = 0**ax**

^{2}+ bx + c = 0⇒ a = 2

⇒ b = 1

⇒ c = 1

Finding out Discriminant of quadratic equation

**2x**

^{2}+ x + 1 = 0D = b

^{2}– 4ac

D = (1)

^{2}– 4 × 2 × 1

D = 1 – 8

**D = -7**

**3. x ^{2} + 3x + 9 = 0**

Comparing

**x**with standard quadratic equation

^{2}+ 3x + 9 = 0**ax**

^{2}+ bx + c = 0⇒ a = 1

⇒ b = 3

⇒ c = 9

Finding out Discriminant of quadratic equation

**x**

^{2}+ 3x + 9 = 0D = b

^{2}– 4ac

D = (3)

^{2}– 4 × 1 × 9

D = 9 – 36

**D = -27**

**4. – x ^{2} + x – 2 = 0**

Comparing

**– x**with standard quadratic equation

^{2}+ x – 2 = 0**ax**

^{2}+ bx + c = 0⇒ a = – 1

⇒ b = 1

⇒ c = – 2

Finding out Discriminant of quadratic equation

**– x**

^{2}+ x – 2 = 0D = b

^{2}– 4ac

D = (1)

^{2}– 4 × (- 1) × (- 2)

D = 1 – 8

**D = – 7**

**5. x ^{2} + 3x + 5 = 0**

Comparing

**x**with standard quadratic equation

^{2}+ 3x + 5 = 0**ax**

^{2}+ bx + c = 0⇒ a = 1

⇒ b = 3

⇒ c = 5

Finding out Discriminant of quadratic equation

**x**

^{2}+ 3x + 5 = 0D = b

^{2}– 4ac

D = (3)

^{2}– 4 × 1 × 5

D = 9 – 20 = – 11

**D = – 11**

**6. x ^{2} – x + 2 = 0**

Comparing

**x**with standard quadratic equation

^{2}– x + 2 = 0**ax**

^{2}+ bx + c = 0⇒ a = 1

⇒ b = – 1

⇒ c = 2

Finding out Discriminant of quadratic equation

**x**

^{2}– x + 2 = 0D = b

^{2}– 4ac

D = (- 1)

^{2}– 4 × 1 × 2

D = 1 – 8 = – 7

**D = – 7**

**8. √3x ^{2} – √2x + 3√3 = 0**

Comparing

**with standard quadratic equation**

**√3x**^{2}– √2x + 3√3 = 0**ax**

^{2}+ bx + c = 0⇒ a = √3

⇒ b = – √2

⇒ c = 3√3

Finding out Discriminant of quadratic equation

**√3x**^{2}– √2x + 3√3 = 0D = b

^{2}– 4ac

D = (√3)

^{2}– 4 × √3 × 3√3

D = 3 – 36 = – 33

**D = – 33**

**9. x ^{2} + x + 1/√2 = 0**

Simplifying Quadratic Equation

**x**

^{2}+ x + 1/√2 = 0**√2x**

^{2}+ √2x + 1 = 0Let’s find out which values of x satisfy this Quadratic Equation

Comparing

**with standard quadratic equation ax**

**√2x**^{2}+ √2x + 1 = 0^{2}+ bx + c = 0

⇒ a = √2

⇒ b = √2

⇒ c = 1

Finding out Discriminant of quadratic equation

**√2x**^{2}+ √2x + 1 = 0D = b

^{2}– 4ac

D = (√2)

^{2}– 4 × √2 × 1

D = 2 – 4√2 = 2(1 – 2√2)

**D = – 2(2√2 – 1)**

**10. x ^{2} + x/√2 + 1 = 0**

Let’s first simplify this Quadratic Equation and then find out values of variable x.

**x**can be rewritten as

^{2}+ x/√2 + 1 = 0**√2x**

^{2}+ x +**√2**= 0Let’s now find out values of x for which Quadratic Equation

**√2x**is satisfied

^{2}+ x +**√2**= 0Comparing

**with standard quadratic equation ax**

**√2x**^{2}+ x +**√2**= 0^{2}+ bx + c = 0

⇒ a = √2

⇒ b = 1

⇒ c = √2

Finding out Discriminant of quadratic equation

**√2x**^{2}+ √2x + 1 = 0D = b

^{2}– 4ac

D = (1)

^{2}– 4 × √2 × √2

D = 1 – 4 × 2 = 1 – 8 = – 7

**D = – 7**