Question

The difference between the roots of the quadratic equation x^2-10x +q=0 is 6. Find q

Answer 1

Answer = 6

Explanation:

I'm guessing there's a simpler solution, but let's give it a shot.

 

The two roots are provided by using the quadratic equation:  

 

x = [-b ± √(b2 - 4ac)] / (2a)

 

For x2 - 10x + q = 0

 

a = 1

b = -10

c = q

 

Substituting these values in:  

 

x = [-b ± √(b2 - 4ac)] / (2a)

  = [-(-10) ± √((-10)2 - 4(1)(q))] / (2(1))

  = [10 ± √(100 - 4q)] / 2

  = 0.5 [10 ± √(100 - 4q)]

 

The first root, x1

 

x1 = 0.5 [10 + √(100 - 4q)]

    = 5 + 0.5√(100 - 4q)

 

The second root, x2

 

x2 = 0.5 [10 - √(100 - 4q)]

    = 5 - 0.5√(100 - 4q)

 

The difference in the roots is 6.

 

6 = x1 - x2  

  = [5 + 0.5√(100 - 4q)] - [5 - 0.5√(100 - 4q)]

  = 5 + 0.5√(100 - 4q) - 5 + 0.5√(100 - 4q)

6 = √(100 - 4q)

 

(6)2 = [√(100 - 4q)]2

36 = 100 - 4q

-64 = -4q

 

q = 16

 

Do a little more work to determine the roots.

 

x2 - 10x + q = 0

x2 - 10x + 16 = 0

(x - 8)(x - 2) = 0

 

x = {2, 8}

 

8 - 2 = 6