Which equation has no real solutions?

3. 4x^2+36 =0 only has a complex solution!

Step-by-step explanation:

1. Solve for x over the real numbers:

4 x^2 + 4 x - 36 = 0

Divide both sides by 4:

x^2 + x - 9 = 0

Add 9 to both sides:

x^2 + x = 9

Add 1/4 to both sides:

x^2 + x + 1/4 = 37/4

Write the left hand side as a square:

(x + 1/2)^2 = 37/4

Take the square root of both sides:

x + 1/2 = sqrt(37)/2 or x + 1/2 = -sqrt(37)/2

Subtract 1/2 from both sides:

x = sqrt(37)/2 - 1/2 or x + 1/2 = -sqrt(37)/2

Subtract 1/2 from both sides:

x = sqrt(37)/2 - 1/2 or x = -1/2 - sqrt(37)/2

2. Solve for x over the real numbers:

4 x^2 - 4 x - 36 = 0

Divide both sides by 4:

x^2 - x - 9 = 0

Add 9 to both sides:

x^2 - x = 9

Add 1/4 to both sides:

x^2 - x + 1/4 = 37/4

Write the left hand side as a square:

(x - 1/2)^2 = 37/4

Take the square root of both sides:

x - 1/2 = sqrt(37)/2 or x - 1/2 = -sqrt(37)/2

Add 1/2 to both sides:

x = 1/2 + sqrt(37)/2 or x - 1/2 = -sqrt(37)/2

Add 1/2 to both sides:

x = 1/2 + sqrt(37)/2 or x = 1/2 - sqrt(37)/2

3. Solve for x:

4 (x^2 + 9) = 0

Divide both sides by 4:

x^2 + 9 = 0

Subtract 9 from both sides:

x^2 = -9

Take the square root of both sides:

x = 3 i or x = -3 i

4. Solve for x over the real numbers:

4 x^2 - 36 = 0

Add 36 to both sides:

4 x^2 = 36

Divide both sides by 4:

x^2 = 9

Take the square root of both sides:

x = 3 or x = -3

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