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

Q: Why do I need to find where the function equals zero first?

The zeros (where $ f(x) = 0 $) are the boundary points where the function changes from positive to negative. These critical points divide the number line into intervals you need to test.

Q: What does the parabola opening upward tell me?

When the coefficient of $ x^2 $ is positive, the parabola opens upward. This means the function will be negative between the roots and positive outside the roots.

Q: Why isn't x = 3 or x = -3 included in the answer?

The inequality is $ f(x) (strictly less than), not \( f(x) ≤ 0 $. At x = ±3, the function equals zero, not less than zero, so we use open intervals.

Q: Can I solve this by graphing instead?

Absolutely! Graph $ y = 3x^2 - 27 $ and look for where the curve is below the x-axis. You'll see it dips below between x = -3 and x = 3.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the following function:

$y=3x^2-27$

Determine for which values of $x$ the following is true:

$f\left(x\right) < 0$

2

Step-by-step solution

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 $-3 < x < 3$ .

3

Final Answer

$-3 < x < 3$

Key Points to Remember

Essential concepts to master this topic

Rule: Find zeros first by setting quadratic equal to zero
Technique: Factor $3x^2 - 27 = 3(x^2 - 9) = 3(x-3)(x+3)$
Check: Test x = 0: $3(0)^2 - 27 = -27 < 0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing inequality direction with parabola shape
Don't think that because the parabola opens upward, the function is always positive = wrong intervals! The parabola being upward means it's negative between the roots and positive outside them. Always test a point in each interval to determine the sign.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why do I need to find where the function equals zero first?

+

The zeros (where $f(x) = 0$ ) are the boundary points where the function changes from positive to negative. These critical points divide the number line into intervals you need to test.

How do I know which interval makes the function negative?

+

Pick any test point in each interval and substitute it into the original function. For $y = 3x^2 - 27$ , try x = 0 (between -3 and 3): $3(0)^2 - 27 = -27$ , which is negative!

What does the parabola opening upward tell me?

+

When the coefficient of $x^2$ is positive, the parabola opens upward. This means the function will be negative between the roots and positive outside the roots.

Why isn't x = 3 or x = -3 included in the answer?

+

The inequality is $f(x) < 0$ (strictly less than), not $f(x) ≤ 0$ . At x = ±3, the function equals zero, not less than zero, so we use open intervals.

Can I solve this by graphing instead?

+

Absolutely! Graph $y = 3x^2 - 27$ and look for where the curve is below the x-axis. You'll see it dips below between x = -3 and x = 3.

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

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to find where the function equals zero first?

How do I know which interval makes the function negative?

What does the parabola opening upward tell me?

Why isn't x = 3 or x = -3 included in the answer?

Can I solve this by graphing instead?

🌟 Unlock Your Math Potential

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