Solve: When is -x² + 2x + 35 Less Than Zero?

To determine where $f(x) < 0$ for the given quadratic function $y = -x^2 + 2x + 35$, we'll perform the following steps:

• Step 1: Identify the roots of the function using the quadratic formula.
• Step 2: Analyze the intervals around the roots to establish where the function is negative.

Step 1: Find the roots using the quadratic formula:

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our function $y = -x^2 + 2x + 35$, we have $a = -1$, $b = 2$, and $c = 35$. Substituting into the formula:

$x = \frac{-2 \pm \sqrt{2^2 - 4(-1)(35)}}{2(-1)}$

$x = \frac{-2 \pm \sqrt{4 + 140}}{-2}$

$x = \frac{-2 \pm \sqrt{144}}{-2}$

$x = \frac{-2 \pm 12}{-2}$

This gives two roots:

- $x_1 = \frac{-2 + 12}{-2} = -5$ - $x_2 = \frac{-2 - 12}{-2} = -7$

Step 2: Analyze the intervals created by the roots:

The roots divide the number line into the intervals $x < -7$, $-7 < x < -5$, and $x > -5$.

Since the parabola $y = -x^2 + 2x + 35$ opens downwards, it will be less than 0 outside the region between the roots. Therefore, the intervals where $y < 0$ are:

• $x > -7$
• $x < -5$

Therefore, the correct answer is:

$x > -7$ or $x < -5$