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Solve the inequality (x - 2)(x + 6) \le -x^2 + 5x - 15. Write your answer in interval notation.
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Starting inequality is
(x - 2)(x + 6) <= -x^2 + 5x - 15.
Simplify and reduce to the standard form quadratic equation
x^2 + 4x - 12 <= -x^2 + 5x - 15,
2x^2 - x + 3 <= 0.
Calculate the discriminant
d = b^2 - 4ac = (-1)^2 - 4*2*3 = 1 - 24 = -23.
The discriminant is negative.
It means that the quadratic function y = 2x^2 - x + 3 has no zeroes and is positive
everywhere on the number line for all real numbers.
Therefore, the set of solutions is EMPTY.
It can not be presented in the interval notation.
Solved.
