Solve Quadratic Inequality: When is x² + 8x - 9 Less Than Zero

To solve the problem, we'll follow these steps:

• Step 1: Use the quadratic formula to find the roots of the equation $x^2 + 8x - 9 = 0$.
• Step 2: Analyze the intervals determined by these roots to find where the function $y = x^2 + 8x - 9$ is less than zero.
• Step 3: Identify the correct inequality and compare with the given multiple-choice options.

Now, let's work through each step:

Step 1: Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 1$, $b = 8$, and $c = -9$:
$x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 + 36}}{2}$ $x = \frac{-8 \pm \sqrt{100}}{2} = \frac{-8 \pm 10}{2}$ This results in the roots $x = 1$ and $x = -9$.

Step 2: Since the quadratic opens upwards (leading coefficient $a = 1$ is positive), the function will be less than zero between the roots. This gives us the interval:
$-9 < x < 1$

Step 3: Identifying the correct choice from the options, the solution is $(-9 < x < 1)$.

Therefore, the solution to the problem, where $y = x^2 + 8x - 9$ is less than zero, is $-9 < x < 1$.