Determine X in the Function y = -x² + 2x + 35 Where f(x) > 0

To solve for when the quadratic function $y = -x^2 + 2x + 35 > 0$, we must first find the roots of the function using the quadratic formula. The quadratic function is given as $y = -x^2 + 2x + 35$.

Step 1: Calculate the roots using the quadratic formula. For $ax^2 + bx + c = 0$, the formula is:

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

Here, $a = -1$, $b = 2$, and $c = 35$. Thus, we compute the discriminant:

$b^2 - 4ac = 2^2 - 4(-1)(35) = 4 + 140 = 144$

Since the discriminant is positive, there are two distinct real roots.

Step 2: Compute the roots using the quadratic formula:

$x = \frac{-2 \pm \sqrt{144}}{2(-1)} = \frac{-2 \pm 12}{-2}$

Calculating the two roots, we get:

$x_1 = \frac{-2 + 12}{-2} = \frac{10}{-2} = -5$ $x_2 = \frac{-2 - 12}{-2} = \frac{-14}{-2} = 7$

Step 3: Determine the intervals where $f(x) > 0$. The roots $x = -5$ and $x = 7$ partition the number line into intervals. A quadratic function with a negative leading coefficient $a$ opens downward, meaning it is positive between its roots:

The intervals are:

• $(-∞, -5)$
• $(-5, 7)$
• $(7, ∞)$

Test the interval between the roots: Choose a point, say $x = 0$, between $-5$ and $7$:

$f(0) = -(0)^2 + 2(0) + 35 = 35 > 0$

This confirms that the function is positive in the interval $(-5, 7)$.

Therefore, the solution to the inequality $f(x) > 0$ is $-5 < x < 7$.

The solution to the problem is $-5 < x < 7$.