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

Q: What does it mean when the discriminant is negative?

When \( \Delta , the quadratic has no real roots. This means the parabola doesn't cross or touch the x-axis at any point.

Q: How do I know if the parabola opens up or down?

Look at the leading coefficient (the number in front of $ x^2 $). If it's positive like a = 1, the parabola opens upward. If negative, it opens downward.

Q: Why can't this function have negative values?

Since the parabola opens upward and never touches the x-axis, it stays completely above the x-axis. The lowest point (vertex) is still positive!

Q: What if the discriminant was positive instead?

If $ \Delta > 0 $, there would be two x-intercepts. Since this parabola opens up, the function would be negative between those two roots.

Q: How can I find the vertex to double-check?

Use $ x = -\frac{b}{2a} = -\frac{8}{2(1)} = -4 $Substitute: $ y = (-4)^2 + 8(-4) + 20 = 16 - 32 + 20 = 4 $Vertex is at (-4, 4), which is above the x-axis!

Quadratic Functions with Negative Value Analysis

Look at the following function:

$y=x^2+8x+20$

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:

$y=x^2+8x+20$

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

$f(x) < 0$

Step-by-step solution

To identify for which values of $x$ the function $y = x^2 + 8x + 20$ is negative, we will analyze the quadratic equation:

Step 1: Identify the coefficients: $a = 1$ , $b = 8$ , $c = 20$ .
Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$ .
Step 3: Evaluate whether the parabola intersects the x-axis based on the discriminant.

Calculating the discriminant:
$\Delta = 8^2 - 4 \cdot 1 \cdot 20 = 64 - 80 = -16$ .

The discriminant $\Delta = -16$ is less than zero, which means there are no real roots. The parabola does not intersect the x-axis and opens upwards because the coefficient a is positive.

Therefore, the values of $y = x^2 + 8x + 20$ are always greater than zero for all real $x$ . The quadratic function does not take negative values for any real $x$ .

The correct answer is: 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: Calculate $\Delta = b^2 - 4ac$ to determine root existence
Analysis Technique: $\Delta = 8^2 - 4(1)(20) = 64 - 80 = -16$
Verification: Since $a > 0$ and $\Delta < 0$ , parabola opens up with no x-intercepts ✓

Common Mistakes

Avoid these frequent errors

Assuming quadratic functions can always be negative
Don't assume every quadratic has negative values just because you're asked to find them = wrong conclusion! When the discriminant is negative and the leading coefficient is positive, the parabola never touches the x-axis and stays entirely above it. Always check the discriminant first to determine if negative values 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 it mean when the discriminant is negative?

When $\Delta < 0$ , the quadratic has no real roots. This means the parabola doesn't cross or touch the x-axis at any point.

How do I know if the parabola opens up or down?

Look at the leading coefficient (the number in front of $x^2$ ). If it's positive like a = 1, the parabola opens upward. If negative, it opens downward.

Why can't this function have negative values?

Since the parabola opens upward and never touches the x-axis, it stays completely above the x-axis. The lowest point (vertex) is still positive!

What if the discriminant was positive instead?

If $\Delta > 0$ , there would be two x-intercepts. Since this parabola opens up, the function would be negative between those two roots.

How can I find the vertex to double-check?

Use $x = -\frac{b}{2a} = -\frac{8}{2(1)} = -4$
Substitute: $y = (-4)^2 + 8(-4) + 20 = 16 - 32 + 20 = 4$
Vertex is at (-4, 4), which is above the x-axis!

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

Quadratic Functions with Negative Value 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 it mean when the discriminant is negative?

How do I know if the parabola opens up or down?

Why can't this function have negative values?

What if the discriminant was positive instead?

How can I find the vertex to double-check?

🌟 Unlock Your Math Potential

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