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

Quadratic Inequalities with Negative Discriminant

Look at the following function:

y=−x2−8x−20 y=-x^2-8x-20

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

f(x)>0 f\left(x\right)>0

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the following function:

y=−x2−8x−20 y=-x^2-8x-20

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

f(x)>0 f\left(x\right)>0

2

Step-by-step solution

To find where the function y=−x2−8x−20 y = -x^2 - 8x - 20 is positive, follow these steps:

Step 1: Calculate the discriminant of the quadratic function.

  • The quadratic discriminant is Δ=b2−4ac \Delta = b^2 - 4ac .
  • In our case, a=−1 a = -1 , b=−8 b = -8 , c=−20 c = -20 , so:
  • Δ=(−8)2−4(−1)(−20)=64−80=−16 \Delta = (-8)^2 - 4(-1)(-20) = 64 - 80 = -16 .

Step 2: Analyze the discriminant.

  • Since Δ<0 \Delta < 0 , the quadratic equation has no real roots.
  • The parabola does not cross the x-axis, and opens downward because a=−1<0 a = -1 < 0 .

Step 3: Determine the sign of the function across the real domain.

  • Without real roots and a parabola opening downwards, the function is negative across its entire domain.

Given this analysis, there are no values of x x for which the function f(x)>0 f(x) > 0 . Therefore, the function has no positive domain.

Conclusion: The function has no positive domain.

3

Final Answer

The function has no positive domain.

Key Points to Remember

Essential concepts to master this topic
  • Discriminant Test: Calculate Δ=b2−4ac \Delta = b^2 - 4ac to find real roots
  • Sign Analysis: When Δ<0 \Delta < 0 and a<0 a < 0 , function is always negative
  • Verification: Test any x-value: f(0)=−20<0 f(0) = -20 < 0 confirms no positive domain ✓

Common Mistakes

Avoid these frequent errors
  • Assuming quadratics always have positive regions
    Don't assume every quadratic has both positive and negative regions = wrong answer! When the discriminant is negative and the parabola opens downward, the function never crosses the x-axis and stays negative everywhere. Always check the discriminant and leading coefficient together.

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
\( f\left(x\right) > 0 \).

AAAX

FAQ

Everything you need to know about this question

What does it mean when the discriminant is negative?

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A negative discriminant means the quadratic has no real roots - the parabola doesn't touch the x-axis at all! This tells us the function stays entirely above or below the x-axis.

How do I know if the function is always positive or always negative?

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Check the leading coefficient (the coefficient of x2 x^2 ). If it's negative like a=−1 a = -1 , the parabola opens downward and stays negative when there are no real roots.

Can I just plug in a test value instead of using the discriminant?

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Yes! You can test any x-value, but using the discriminant first gives you the complete picture. For example, f(0)=−20 f(0) = -20 shows the function is negative at zero, confirming our analysis.

Why can't this function ever be positive?

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Since the parabola opens downward (because a=−1<0 a = -1 < 0 ) and never crosses the x-axis (because Δ=−16<0 \Delta = -16 < 0 ), it stays below the x-axis for all real numbers.

What would happen if the leading coefficient were positive instead?

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If a>0 a > 0 with the same negative discriminant, the parabola would open upward and never cross the x-axis, meaning the function would be always positive instead!

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