To solve the inequality $x(x + 9) + 24 > 17$, follow these steps:

1. Simplify the inequality: $x(x + 9) + 24 > 17$ First, expand the expression: $x^2 + 9x + 24 > 17$

2. Move all terms to one side: Subtract 17 from both sides: $x^2 + 9x + 24 - 17 > 0$ Simplify: $x^2 + 9x + 7 > 0$

3. Solve the corresponding quadratic equation: Solve $x^2 + 9x + 7 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 9$, and $c = 7$.

   Calculate the discriminant: $\text{Discriminant} = b^2 - 4ac = 9^2 - 4(1)(7) = 81 - 28 = 53$ Since the discriminant is positive, there are two real roots: $x = \frac{-9 \pm \sqrt{53}}{2}$

4. Find the roots: $x_1 = \frac{-9 + \sqrt{53}}{2}, \quad x_2 = \frac{-9 - \sqrt{53}}{2}$ These are the roots of the quadratic equation.

5. Determine the intervals where the inequality holds: The quadratic $x^2 + 9x + 7$ opens upwards (since the coefficient of $x^2$ is positive), so the inequality $x^2 + 9x + 7 > 0$ holds for $x$ values outside the interval defined by the roots. Therefore, the solution to the inequality is: $x < \frac{-9 - \sqrt{53}}{2} \quad \text{or} \quad x > \frac{-9 + \sqrt{53}}{2}$

Final Answer:

$x < \frac{-9 - \sqrt{53}}{2} \quad \text{or} \quad x > \frac{-9 + \sqrt{53}}{2}$

Would you like more details on any of these steps, or do you have any questions?

Related Questions:

1. How do you solve a quadratic equation by factoring?
2. What are the properties of the discriminant in the quadratic formula?
3. Can a quadratic inequality ever have no real solutions?
4. How does the sign of the leading coefficient in a quadratic equation affect the graph?
5. What are the methods to solve a quadratic inequality?

Tip:

When solving inequalities, always remember to consider the sign of the quadratic expression and the intervals between the roots.