Question

Use the discriminant to describe the roots of each equation. Then select the best description?

x^2-5x-4=0

double root
Real and irrational root
Real and rational root
Imaginary root

Answer 1

b

Explanation:

Calculate the value of the discriminant

Δ = b² - 4ac

• If b² - 4ac > 0 then roots are real and irrational

• If b² - 4ac > 0 and a perfect square, roots are real and rational

• If b² - 4ac = 0 then roots are equal, double root

• If b² - 4ac < 0 then roots are not real, imaginary roots

For x² - 5x - 4 = 0

with a = 1, b = - 5 and c = - 4, then

b² - 4ac

= (- 5)² - (4 × 1 × - 4)

= 25 + 16

= 41

Since b² - 4ac > 0 then roots are real and irrational

Answer 2

The roots are real and irrational

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