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10 October, 13:27

A) One of the roots of the equation 10x^2-33x+c=0 is 5.3. Find the other root and the coefficient c.

b) The difference between the roots of the quadratic equation x^2-12x+q=0 is 2. Find q.

c) The difference between the roots of the quadratic equation x^2+x+c=0 is 6. Find c.

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  1. 10 October, 14:08
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    a) The other root is - 2

    The coefficient c = - 106

    b) q = 35

    c) c = - 8.75

    Step-by-step explanation:

    * Lets study the general form of the quadratic equation

    * ax² + bx + c = 0

    - Their roots are x1 and and x2

    - The sum of them = - b/a ⇒ x1 + x2 = - b/a

    - The product of them = c/a ⇒ (x1) (x2) = c/a

    a) * Assume that the roots of the equation 10x² - 33x + c = 0

    are m and n

    ∵ m + n = - b/a

    ∵ a = 10 and b = - 33

    ∴ m + n = - (-33) / 10 = 3.3

    ∵ m = 5.3

    ∴ 5.3 + n = 3.3 ⇒ n = 3.3 - 5.5 = - 2

    ∴ n = - 2

    * The other root is - 2

    ∵ m * n = c/a

    ∵ m = 5.3, n = - 2, a = 10

    ∴ (5.3) (-2) = c/10

    ∴ - 10.6 = c/10 ⇒ Multiply both sides by 10

    ∴ c = - 106

    * The coefficient c = - 106

    b) * Assume that the roots of the equation x² - 12x + q = 0

    are m and n

    ∵ The difference between the roots is 2

    ∴ m - n = 2 ⇒ (1)

    ∵ From the equation m + n = - b/a

    ∵ a = 1, b = - 12

    ∴ m + n = - (-12) / 1 = 12

    ∴ m + n = 12 ⇒ (2)

    * Lets solve the two equation

    - Add the two equation to eliminate n

    ∴ 2m = 14 ⇒ divide both sides by 2

    ∴ m = 7

    * Substitute the value of m in (1) or (2)

    ∴ 7 - n = 2 ⇒ 7 - 2 = n ⇒ 5 = n

    ∴ n = 5

    ∵ mn = c/a

    ∵ c = q, a = 1

    ∴ mn = q/1 = q

    ∴ q = 7 * 5 = 35

    * q = 35

    c) * Assume that the roots of the equation x² + x + c = 0

    are m and n

    ∵ The difference between the roots is 6

    ∴ m - n = 6 ⇒ (1)

    ∵ From the equation m + n = - b/a

    ∵ a = 1, b = 1

    ∴ m + n = - (1) / 1 = - 1

    ∴ m + n = - 1 ⇒ (2)

    * Lets solve the two equation

    - Add the two equation to eliminate n

    ∴ 2m = 5 ⇒ divide both sides by 2

    ∴ m = 2.5

    * Substitute the value of m in (1) or (2)

    ∴ 2.5 + n = - 1 ⇒ n = - 1 - 2.5 = - 3.5

    ∴ n = - 3.5

    ∵ mn = c/a

    ∵ c = c, a = 1

    ∴ mn = c/1 = c

    ∴ c = 2.5 * (-3.5) = - 8.75

    * c = - 8.75
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