Answer:
The roots are real and irrational
Step-by-step explanation:
* Lets explain what is the discriminant
- In the quadratic equation ax² + bx + c = 0, the roots of the
equation has three cases:
1- Two different real roots
2- One real root or two equal real roots
3- No real roots means imaginary roots
- All of these cases depend on the value of a , b , c
∵ The rule of the finding the roots is
x = [-b ± √(b² - 4ac)]/2a
- The effective term is √(b² - 4ac) to tell us what is the types of the root
# If the value under the root b² - 4ac positive means greater than 0
∴ There are two different real roots
# If the value under the root b² - 4ac = 0
∴ There are two equal real roots means one real root
# If the value under the root b² - 4ac negative means smaller than 0
∴ There is real roots but the roots will be imaginary roots
∴ We use the discriminant to describe the roots
* Lets use it to check the roots of our problem
∵ x² - 5x - 4 = 0
∴ a = 1 , b = -5 , c = -4
∵ Δ = b² - 4ac
∴ Δ = (-5)² - 4(1)(-4) = 25 + 16 = 41
∵ 41 > 0
∴ The roots of the equation are two different real roots
∵ √41 is irrational number
∴ The roots are real and irrational
* Lets check that by solving the equation
∵ x = [-(-5) ± √41]/2(1) = [5 ± √41]/2
∴ x = [5+√41]/2 , x = [5-√41]/2 ⇒ both real and irrational
