Solve for Positive Values: y = 2x² - 4x + 5

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

A negative discriminant means the parabola never crosses the x-axis. It has no real roots, so the function never equals zero. This is key information for determining where the function is positive or negative!

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 positive, the parabola opens upward. Combined with no x-intercepts, this means the function is always above the x-axis!

Q: What if the discriminant were positive instead?

A positive discriminant means the parabola crosses the x-axis at two points. The function would be positive in some intervals and negative in others, depending on which side of the roots you're on.

Q: Why can't I just look at the vertex?

The vertex tells you the minimum or maximum value, but not whether the function crosses zero. For $ y = 2x^2 - 4x + 5 $, the vertex is at $ y = 3 $, which is positive, confirming our answer!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the following function:

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

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

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

2

Step-by-step solution

To determine for which values of $x$ the function $y = 2x^2 - 4x + 5$ is positive, we will analyze its characteristics.

Step 1: Determine the direction of the parabola.
The given quadratic function $y = 2x^2 - 4x + 5$ has a leading coefficient $a = 2$ , which is positive. Therefore, the parabola opens upwards.

Step 2: Check for real roots.
To identify where the function might be zero, calculate the discriminant $\Delta = b^2 - 4ac$ .
Here, $a = 2$ , $b = -4$ , $c = 5$ .
The discriminant $\Delta = (-4)^2 - 4 \cdot 2 \cdot 5 = 16 - 40 = -24$ .
Since the discriminant is negative, the quadratic has no real roots, meaning it doesn't intersect the x-axis.

Step 3: Analyze positivity over the entire domain.
Since the parabola opens upwards and has no real roots, the function does not touch or cross the x-axis. Therefore, $y = 2x^2 - 4x + 5$ is always positive.

Conclusion.
The function $y = 2x^2 - 4x + 5$ is positive for all values of $x$ .

Therefore, the solution to the problem is The function is positive for all values of $x$ .

3

Final Answer

The function is positive for all values of $x$ .

Key Points to Remember

Essential concepts to master this topic

Direction: Positive leading coefficient means parabola opens upwards
Discriminant: Calculate $\Delta = b^2 - 4ac = (-4)^2 - 4(2)(5) = -24$
Check: Negative discriminant + upward parabola = always positive ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check the parabola's direction
Don't just calculate the discriminant and assume the function changes sign! A negative discriminant only tells you there are no x-intercepts. Always check if the leading coefficient is positive (opens up) or negative (opens down) to determine if the function is always positive or always negative.

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 crosses the x-axis. It has no real roots, so the function never equals zero. This is key information for determining where the function is positive or negative!

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 positive, the parabola opens upward. Combined with no x-intercepts, this means the function is always above the x-axis!

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

+

You could test one value, but the discriminant method is more reliable and complete. It tells you definitively whether the parabola crosses the x-axis, while testing just gives you one point.

What if the discriminant were positive instead?

+

A positive discriminant means the parabola crosses the x-axis at two points. The function would be positive in some intervals and negative in others, depending on which side of the roots you're on.

Why can't I just look at the vertex?

+

The vertex tells you the minimum or maximum value, but not whether the function crosses zero. For $y = 2x^2 - 4x + 5$ , the vertex is at $y = 3$ , which is positive, confirming our answer!

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Solve for Positive Values: y = 2x² - 4x + 5

Quadratic Functions with Discriminant Analysis

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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 a test value instead of using the discriminant?

What if the discriminant were positive instead?

Why can't I just look at the vertex?

🌟 Unlock Your Math Potential

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