Answer:
q = 91
Step-by-step explanation:
We are going to use formula for finding roots to solve this questions;
Using this two formula;
ax² + bx + c = 0---------(1)
Let ∝ and β be the two roots of the quadratic equation above, then
x² - (∝ + β)x + ∝β = 0 --------(2)
comparing the two equations above
sum of the roots of a quadratic equation is : ∝ + β = -b/a -----(3)
Products of the root of a quadratic equation is : ∝β = c/a ------(4)
We can now start solving our question;
<u><em>The difference between the roots of the quadratic equation x^2−20x+q=0 is 6. Find q.</em></u>
Let the two roots be ∝ and ∝ - 6
Sum of the roots = ∝ + ∝ - 6 = 2∝ - 6 -----------(5)
Product of the roots = ∝( ∝ - 6) = ∝² - 6∝ ------(6)
From equation (3) and equation (4)
sum of a quadratic equation = -b/a
Product of a quadratic equation = c/a
so;
From equation (5) and (6)
Sum of the roots = 2∝ - 6 = -b/a ---------(7)
Product of the roots = ∝² - 6∝ = c/a --------(8)
But again the equation given to us is;
x²−20x+q=0
comparing this with the standard equation, equation(1)
a = 1 b = 20 and c = q
From equation (7) : 2∝ - 6 = -b/a
2∝ - 6 = -20/1
2∝ - 6 = -20
Add 6 to both-side of the equation
2∝ - 6 + 6 = -20+ 6
2∝= -14
Divide both-side of the equation by 2
∝ = -7
Also, from equation (8) : ∝² - 6∝ = c/a
∝² - 6∝ = q/1
∝² - 6∝ = q
But ∝ = -7, so we will substitute ∝ = -7 in the above equation to get the value of q;
∝² - 6∝ = q
(-7)² - 6(-7) = q
49 + 42 = q
91 = q
Therefore the value of q = 91
[To test the correctness of our answer, lets substitute q back into the equation and solve;
x²−20x+91=0
x² - 7x - 13x + 91 = 0
x(x-7) -13(x-7) = 0
(x-7)(x-13)= 0
x = 7 or x=13
The difference between 7 and 13 is 6.]
Therefore the value of q is 91
