How many real number solutions are there to the equation 0 = -3x² + x - 4

Solution:

Given equation is -3x² + x - 4 = 0

Discriminant determines the nature of the quadratic solution

It is calculated as Δ = b² - 4ac

In the given equation, a = -3, b =1 and c = -4

= (1)² - 4(-3)(-4)

= 1 - 48 = -47

⇒ Negative value

Therefore, no real solutions.

Note: If quadratic equation is ax² + bx + c = 0, then discriminant Δ = b² - 4ac decides nature of roots of quadratic equation,

(i) If Δ = 0, then roots are real and repeat

(ii) If Δ > 0, then roots are real and distinct

(iii) If Δ < 0, then roots are imaginary or complex or non-real

How many real number solutions are there to the equation 0 = -3x² + x - 4

Summary:

There are no real solutions for the given equation 0 = -3x² + x - 4.