Finding Values Where y=3x²-27 Falls Below Zero

To solve the problem of finding the values of $x$ for which $y = 3x^2 - 27 < 0$, we start by solving the equation $3x^2 - 27 = 0$:

Step 1: Solve the equation $3x^2 - 27 = 0$.

• Rearrange the equation as $3x^2 = 27$.
• Divide both sides by 3 to simplify: $x^2 = 9$.
• Take the square root of both sides: $x = \pm 3$.

The solutions $x = 3$ and $x = -3$ are the roots of the quadratic function. This means the function transitions from negative to non-negative (and vice versa) at these points.

Step 2: Analyze the intervals defined by the roots.

• The potential intervals of interest are $x < -3$, $-3 < x < 3$, and $x > 3$.

Since the quadratic is a parabola opening upwards (coefficient of $x^2$ is positive), the function $y = 3x^2 - 27$ will be negative between the roots.

Therefore, check the interval $-3 < x < 3$:

• When $x = 0$, substitute into the function: $y = 3(0)^2 - 27 = -27$, which is negative.

The function $y = 3x^2 - 27$ is negative in the interval $-3 < x < 3$.

Thus, the values of $x$ for which $f(x) < 0$ are $-3 < x < 3$.