Determine Critical Values for y = 3x² - 6x + 4: When is f(x) < 0?

Q: What does a negative discriminant actually mean?

A negative discriminant means the parabola never touches the x-axis. Since our coefficient $ a = 3 > 0 $, the parabola opens upward and stays completely above the x-axis.

Q: How can I be sure the function is never negative?

Two key facts: (1) $ \Delta = -12 means no x-intercepts, and (2) \( a = 3 > 0 $ means upward opening. Together, these guarantee the function is always positive.

Q: What if the coefficient of x² was negative instead?

If \( a with \( \Delta , the parabola would open downward and stay completely below the x-axis, making the function always negative.

Q: Do I need to find the vertex to solve this?

Not necessarily! The discriminant test is faster. Once you know $ \Delta and \( a > 0 $, you immediately know the function is always positive.

Q: Can I test specific values instead of using the discriminant?

You could test values, but that's not a proof. The discriminant method gives you a complete mathematical proof that covers all possible x-values, not just the ones you test.

Quadratic Functions with Discriminant Analysis

Look at the following function:

$y=3x^2-6x+4$

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

$f(x) < 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$y=3x^2-6x+4$

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

$f(x) < 0$

Step-by-step solution

Let's analyze the function $y = 3x^2 - 6x + 4$ to determine when it is negative.

First, calculate the discriminant $\Delta$ :

$\Delta = (-6)^2 - 4 \cdot 3 \cdot 4 = 36 - 48 = -12$

A negative discriminant ( $\Delta < 0$ ) indicates that there are no real roots, meaning the graph of the function does not intersect the x-axis. Since $a = 3 > 0$ , the parabola opens upwards.

This means the vertex of the parabola represents the minimum point, and the entire graph is above the x-axis.

Consequently, the function $y = 3x^2 - 6x + 4$ does not attain any negative values for any real $x$ .

The correct interpretation is that the function stays positive, confirming the conclusion:

The function has no negative values.

Final Answer

The function has no negative values.

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $\Delta < 0$ , quadratic has no real roots
Sign Analysis: If $a > 0$ and $\Delta < 0$ , function is always positive
Check: Calculate $\Delta = b^2 - 4ac = 36 - 48 = -12$ ✓

Common Mistakes

Avoid these frequent errors

Trying to find x-intercepts when discriminant is negative
Don't attempt to solve $3x^2 - 6x + 4 = 0$ when $\Delta = -12 < 0$ = no real solutions exist! This wastes time and creates confusion. Always check the discriminant first to determine if real roots exist.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

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

FAQ

Everything you need to know about this question

What does a negative discriminant actually mean?

A negative discriminant means the parabola never touches the x-axis. Since our coefficient $a = 3 > 0$ , the parabola opens upward and stays completely above the x-axis.

How can I be sure the function is never negative?

Two key facts: (1) $\Delta = -12 < 0$ means no x-intercepts, and (2) $a = 3 > 0$ means upward opening. Together, these guarantee the function is always positive.

What if the coefficient of x² was negative instead?

If $a < 0$ with $\Delta < 0$ , the parabola would open downward and stay completely below the x-axis, making the function always negative.

Do I need to find the vertex to solve this?

Not necessarily! The discriminant test is faster. Once you know $\Delta < 0$ and $a > 0$ , you immediately know the function is always positive.

Can I test specific values instead of using the discriminant?

You could test values, but that's not a proof. The discriminant method gives you a complete mathematical proof that covers all possible x-values, not just the ones you test.

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Determine Critical Values for y = 3x² - 6x + 4: When is f(x) < 0?

Quadratic Functions with Discriminant Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What does a negative discriminant actually mean?

How can I be sure the function is never negative?

What if the coefficient of x² was negative instead?

Do I need to find the vertex to solve this?

Can I test specific values instead of using the discriminant?

🌟 Unlock Your Math Potential

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Positive and Negative Domains

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Solve y = x² + 8x + 20: Finding Values Where Function is Positive