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

AAABBBCCCX

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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