Identify When y = -x² - 2x - 3 Surpasses Zero: Solving the Quadratic Inequality

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

A negative discriminant means the quadratic has no real roots - the parabola never touches or crosses the x-axis. This is actually quite common and completely normal!

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

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

Q: Can I just plug in any number to check the sign?

Yes! Since the function never changes sign (no roots), testing any value tells you the sign everywhere. Try $ x = 0 $: \( f(0) = -3 .

Q: Why can't this function ever be positive?

Think of it visually: this parabola opens downward (like an upside-down U) and floats entirely below the x-axis. It never gets high enough to be positive!

Q: What if the question asked for f(x) ≥ 0 instead?

The answer would still be no solution because this function is never zero or positive. It's always strictly negative for all real numbers.

Quadratic Inequalities with Negative Discriminants

Look at the following function:

$y=-x^2-2x-3$

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

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

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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:

$y=-x^2-2x-3$

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

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

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Find the roots of the function using the quadratic formula. The equation to solve is $-x^2 - 2x - 3 = 0$ .
Step 2: The quadratic formula gives $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . For our function, $a = -1$ , $b = -2$ , and $c = -3$ .
Step 3: Calculate the discriminant: $b^2 - 4ac = (-2)^2 - 4(-1)(-3) = 4 - 12 = -8$ .
Step 4: Since the discriminant is negative, there are no real roots. Therefore, $f(x)$ never crosses the x-axis and has no real x-intercepts.
Step 5: Understand the graph behavior: since the parabola opens downward and the vertex is its maximum point without crossing the x-axis, the function is negative for all real $x$ .

Therefore, the solution to the problem is that the function has no positive domain.

Final Answer

The function has no positive domain.

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $b^2 - 4ac < 0$ , no real roots exist
Technique: Check parabola direction: $a = -1 < 0$ means opens downward
Check: Test any x-value: $f(0) = -3 < 0$ , confirms always negative ✓

Common Mistakes

Avoid these frequent errors

Assuming the function has roots when discriminant is negative
Don't try to find real x-intercepts when discriminant = -8 < 0, leading to impossible square roots! This creates confusion about where the function changes sign. Always recognize that negative discriminant means the parabola never crosses the x-axis.

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

What does it mean when the discriminant is negative?

A negative discriminant means the quadratic has no real roots - the parabola never touches or crosses the x-axis. This is actually quite common and completely normal!

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

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

Can I just plug in any number to check the sign?

Yes! Since the function never changes sign (no roots), testing any value tells you the sign everywhere. Try $x = 0$ : $f(0) = -3 < 0$ .

Why can't this function ever be positive?

Think of it visually: this parabola opens downward (like an upside-down U) and floats entirely below the x-axis. It never gets high enough to be positive!

What if the question asked for f(x) ≥ 0 instead?

The answer would still be no solution because this function is never zero or positive. It's always strictly negative for all real numbers.

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Identify When y = -x² - 2x - 3 Surpasses Zero: Solving the Quadratic Inequality

Quadratic Inequalities with Negative Discriminants

❤️ 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 function is always positive or always negative?

Can I just plug in any number to check the sign?

Why can't this function ever be positive?

What if the question asked for f(x) ≥ 0 instead?

🌟 Unlock Your Math Potential

More Questions

Standard Representation

Solve y = -x² - 8x - 20: Finding Where Function Values Are Negative

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Determining Negative Regions of the Quadratic: y = -2x² - 8x - 10

Positive and Negative Domains

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

Finding X: Solve y = x² + 5x + 4 for Negative Output

Solve y=-x²-2x-3: Finding Where Function Values Are Negative

Determine x: Solving y = -2x² + 8x - 10 for Positive Values