Find all positive integer values of c such that the equation x^2-7x+c=0 only has roots that are real and rational.
To find all positive integer values of c such that the equation x^2-7x+c=0 only has roots that are real and rational, we can use the discriminant of the quadratic equation.
The discriminant (denoted by Δ) of a quadratic equation ax^2 + bx + c = 0 can be found using the formula Δ = b^2 - 4ac.
For the equation x^2 - 7x + c = 0, the coefficient of the quadratic term (a) is 1, the coefficient of the linear term (b) is -7, and the constant term (c) is c.
Since we want the roots to be real and rational, the discriminant (Δ) should be a perfect square.
Therefore, we have the inequality Δ = (-7)^2 - 4(1)(c) ≥ 0.
Simplifying this inequality gives 49 - 4c ≥ 0.
Solving this inequality gives c ≤ 12.
Since we are looking for positive integer values of c, the possible solutions are 1, 2, 3, ..., 12.
Hence, the positive integer values of c for which the equation x^2 - 7x + c = 0 only has roots that are real and rational are 1, 2, 3, ..., 12.
the two roots have product c and sum 7
Since the coefficient of x is 1, all the rational roots are integers.
(x-1)(x-6) = x^2-7x+6
(x-2)(x-5) = x^2-7x+10
...
I think you can see that there are only a few values for c.