Question

Analizando el discriminante indica si las siguientes ecuaciones cuadráticas tienen una solución, dos soluciones o no tienen solución en números reales.

a) x – 3x2 + 9=0
b) 2x2 + x - 1=0
c) – 3x2 + x - 10=0

Answer 1

Rewrite the equation in the canonic order
`ax^2+bx+c=0`:

• 
  `-3x^2+x+9 = 0`
• 
  `2x^2+x-1 = 0`
• 
  `-3x^2 + x - 10 = 0`

The discriminant is defined as
`\Delta = b^2-4ac`. There are three cases:

• 
  `\Delta > 0 \to \text{two different solutions}`
• 
  `\Delta = 0 \to \text{one solution}`
• 
  `\Delta < 0 \to \text{no solutions}`

Let's compute the discriminant for each equation:

• 
  `-3x^2+x+9 = 0 \to 1-4\cdot(-3)\cdot 9 = 1 + 108 = 109 \to \text{two different solutions}`
• 
  `2x^2+x-1 = 0 \to 1-4\cdot 2\cdot (-1) = 1+8=9 \to \text{two different solutions}`
• 
  `-3x^2 + x - 10 = 0 \to 1-4\cdot (-3) \cdot (-10) 1-120 = -119 \to \text{no solutions}`