Solve the Quadratic Inequality: When is -x² + 4x - 5 < 0?

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

A negative discriminant means the parabola never touches or crosses the x-axis. Since our parabola $ y = -x^2 + 4x - 5 $ opens downward (a = -1), it stays completely below the x-axis.

Q: How do I know which direction the parabola opens?

Look at the coefficient of $ x^2 $! If a > 0, it opens upward (U-shape). If a , it opens downward (∩-shape). Here, a = -1

Q: Why is the function negative for ALL values of x?

Since the downward-opening parabola never crosses the x-axis, it lies entirely below the x-axis. This means every point on the curve has a negative y-value.

Q: Can I just plug in numbers to check?

Yes! Try any x-value: x = 0: \( -0^2 + 4(0) - 5 = -5 x = 2: \( -(2)^2 + 4(2) - 5 = -1 Every value gives a negative result!

Q: What if the parabola opened upward instead?

If a > 0 and discriminant entirely above the x-axis, making the function always positive instead of always negative.

Quadratic Inequalities with Negative Discriminant

Look at the following function:

$y=-x^2+4x-5$

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:

$y=-x^2+4x-5$

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

$f(x) < 0$

Step-by-step solution

Let's analyze the function $y = -x^2 + 4x - 5$ and determine the interval where $y$ is negative.

1. **Find the roots using the quadratic formula**:
The function is given by $y = -x^2 + 4x - 5$ . The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = -1$ , $b = 4$ , and $c = -5$ . First, we calculate the discriminant:

b^2 - 4ac = 4^2 - 4(-1)(-5) = 16 - 20 = -4

Since the discriminant is negative, the quadratic equation has no real roots, implying that the parabola does not intersect the x-axis. The quadratic formula confirms there are no real solutions, confirming the function does not touch or cross the x-axis.

2. **Analyze the parabola's direction**:
Since $a = -1$ , the parabola opens downwards. A downward-opening parabola with no real roots means it lies entirely below the x-axis. Hence, the function $y = -x^2 + 4x - 5$ is negative for all values of $x$ .

Therefore, the function is negative for all $x$ .

Final Answer

The function is negative for all $x$ .

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When b² - 4ac < 0, parabola never crosses x-axis
Direction Analysis: Since a = -1 < 0, parabola opens downward completely
Verification: Test any x-value: (-1)² + 4(1) - 5 = -2 < 0 ✓

Common Mistakes

Avoid these frequent errors

Trying to solve by setting the quadratic equal to zero
Don't find roots when discriminant is negative = no real solutions exist! This wastes time and confuses the inequality analysis. Always check the discriminant first: $b^2 - 4ac = 16 - 20 = -4 < 0$ means no x-intercepts.

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 parabola never touches or crosses the x-axis. Since our parabola $y = -x^2 + 4x - 5$ opens downward (a = -1), it stays completely below the x-axis.

How do I know which direction the parabola opens?

Look at the coefficient of $x^2$ ! If a > 0, it opens upward (U-shape). If a < 0, it opens downward (∩-shape). Here, a = -1 < 0, so it opens downward.

Why is the function negative for ALL values of x?

Since the downward-opening parabola never crosses the x-axis, it lies entirely below the x-axis. This means every point on the curve has a negative y-value.

Can I just plug in numbers to check?

Yes! Try any x-value:

x = 0: $-0^2 + 4(0) - 5 = -5 < 0$
x = 2: $-(2)^2 + 4(2) - 5 = -1 < 0$

Every value gives a negative result!

What if the parabola opened upward instead?

If a > 0 and discriminant < 0, the upward-opening parabola would lie entirely above the x-axis, making the function always positive instead of always negative.

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Solve the Quadratic Inequality: When is -x² + 4x - 5 < 0?

Quadratic Inequalities with Negative Discriminant

❤️ 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 which direction the parabola opens?

Why is the function negative for ALL values of x?

Can I just plug in numbers to check?

What if the parabola opened upward instead?

🌟 Unlock Your Math Potential

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