quadratic Equations

Question

Find the value of q, for which the difference between the roots in the equation x^2−10x+q=0 is equal to 6.

Find the value of q, for which the difference between the roots in the equation x²−10x+q=0 is equal to 6.

Answer 1

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