Answer:
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
