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3 years ago

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

Mathematics

1 answer:

notsponge [240]3 years ago

5 0

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

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