In complete sentences, explain why 1/(x^2+2x+9) <0 has no solution. Do NOT solve the inequality.
for 1/y < 0 , y has to be negative
so look at the graph of y = x^2 + 2x + 9
the x of its vertex is -b/2a = -2/2 = -1, and
the y of the vertex is (-1)^2 - 2 + 9 = 8
the vertex is (-1,8), which lies above the x-axis
and of course our parabola opens upwards, so it cannot go below the x-axis
To determine why the inequality 1/(x^2+2x+9) < 0 has no solution, we need to analyze the quadratic expression x^2+2x+9.
1. Identify the discriminant: The discriminant of the quadratic expression ax^2+bx+c is given by Delta = b^2 - 4ac. In our case, a = 1, b = 2, and c = 9. Substituting these values, we find the discriminant as Delta = 2^2 - 4(1)(9) = 4 - 36 = -32.
2. Interpret the discriminant: The discriminant tells us about the nature of the roots of a quadratic equation. When the discriminant is negative (as in this case, -32 < 0), it means that the quadratic equation will have no real roots.
3. Relate to the inequality: Since 1/(x^2+2x+9) is a rational function, it consists of a fraction. To satisfy the inequality 1/(x^2+2x+9) < 0, the function must be negative; in other words, the numerator must be negative, and the denominator must be positive.
However, since the quadratic expression x^2+2x+9 has no real roots due to the negative discriminant, it means that it is always positive or always negative. As such, there are no values of x for which the inequality 1/(x^2+2x+9) < 0 is satisfied, thereby resulting in no solution.