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25 March, 23:53

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

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Answers (2)
  1. 26 March, 00:05
    0
    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
  2. 26 March, 00:19
    0
    b

    Step-by-step 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
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