The difference between the roots of the quadratic equation x2−12x+q=0 is 2. Find q.

Question

The difference between the roots of the quadratic equation x2−12x+q=0 is 2. Find q.

The difference between the roots of the quadratic equation x²−12x+q=0 is 2. Find q.

Answer 1

x² - 12x + q = 0

 

Let the two roots be r and r+2.

 

Factor the quadratic expression:

 

(x - r)[x - (r + 2)] = 0

 

Expand, simplify, group like terms, and get

 

x² - 2(r + 1)x + r(r + 2) = 0

 

Compare to 

 

x² - 12x + q = 0

 

and set equal the coefficients of like terms:

 

Coefficient of x:

-2(r + 1) = -12 ⇒ r + 1 = 6 ⇒ r = 5

(Then the other root is r + 2 = 5 + 2 = 7)

 

Constant term:

r(r + 2) = q ⇒ 5(5 + 2) = q

 

q = 35

 

Test the solution:

 

(x - 5)(x - 7) = x² - 12x + 35

 

With two roots differing by 2, you get an equation of the form 

 

x² - 12x + q = 0

 

with q = 35.