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.

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