Solution:
Given that equation is:
[tex]x^2 - kx + 28 = 0[/tex]
Roots are:
[tex]\alpha\\\\\alpha + 3[/tex]
To find: value of x
The general quadratic equation is:
[tex]ax^2 + bx + c = 0[/tex]
[tex]\text{ Product of roots } = \frac{c}{a}\\\\\text{ Sum of roots } = \frac{-b}{a}[/tex]
From given,
[tex]x^2 - kx + 28 = 0[/tex]
a = 1
b = -k
c = 28
Therefore,
[tex]Product\ of\ roots = \frac{28}{1} = 28[/tex]
[tex]Sum\ of\ roots = \frac{-k}{1} = -k[/tex]
Given roots are:
[tex]\alpha\\\\\alpha + 3[/tex]
Therefore,
The two roots are two numbers whose difference is 3 and whose product is 28
Those two roots are 4 and 7 or -4 and -7
Then, sum of roots are:
4 + 7 = 11
-4 - 7 = -11
Therefore, the possible values of k are -11 and 11
