Question

How do you solve `x^2 - 12x +35 ≤ 0`?

Answer 1

Graph and solve the inequality in one variable: `f(x) = x^2 - 12x + 35 <= 0`

Explanation:

First, solve f(x) = 0 to find the 2 x-intercepts (real roots)
`f(x) = x^2 - 12x + 35 = 0`
The roots have same sign. Factor pairs of 35 --> (1, 35)(5, 7). This sum is 12 = -b. Then, the 2 real roots are 5 and 7.
Since a = 1 > 0, the parabola opens upward. Between the 2 x-intercepts, the graph is below the x axis, meaning f(x) < 0 in this interval (5, 7).
The solution set is the closed interval: [5, 7]. The 2 critical points 5 and 7 are included in the solution set
graph{x^2 - 12x + 35 [-10, 10, -5, 5]}