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Answer: k=12
Step-by-step explanation:
x2 - 7x + k = 0
using quadratic formula method,
x = [-b ± √(b2 - 4ac)]/2a
a = 1
b = -7
c = k
x = [-(-7) ± √((-7)2 - 4(1)(k))]/2(1)
x = [7 ± √(49 - 4k)]/2
x = [7 + √(49 - 4k)] / 2 and [7 - √(49 - 4k)] / 2
since the roots are m and m-1,
m = [7 + √(49 - 4k)] / 2 ------------ eqn(1), and
m - 1 = [7 - √(49 - 4k)] / 2 ----------- eqn(2)
add 1 to both sides of eqn(2)
m = [7 - √(49 - 4k)]/2 + 1 --------------- eqn(3)
eqn(1) = eqn(3)
[7 + √(49 - 4k)]/2 = [7 - √(49 - 4k)]/2 + 1
7/2 + (√(49 - 4k))/2 = 7/2 - (√(49 - 4k))/2 + 1
collect like terms,
(√(49 - 4k))/2 + (√(49 - 4k))/2 = 7/2 - 7/2 + 1
(√(49 - 4k))/2 + (√(49 - 4k))/2 = 0 + 1
(√(49 - 4k))/2 + (√(49 - 4k))/2 = 1
multiply through by 2
√(49 - 4k) + √(49 - 4k) = 2
2√(49 - 4k) = 2
divide both side by 2
√(49 - 4k) = 1
square both sides
49 - 4k = 1
collect like terms,
4k = 49 - 1
4k = 48
divide both sides by 4
k = 48/4
k = 12