Solve the Quadratic: Discover X in y=2x^2-4x+5 Where f(x) < 0

Q: Why can't I just solve the equation by setting it equal to zero?

Setting the quadratic equal to zero finds roots (where it crosses the x-axis). But we need to know where it's negative, which requires checking the vertex and parabola direction first!

Q: How do I know if a parabola opens upward or downward?

Look at the leading coefficient (the number in front of $ x^2 $). If it's positive, the parabola opens upward. If negative, it opens downward.

Q: What if the vertex is negative but the parabola opens upward?

Then the function would have negative values! The parabola would dip below the x-axis around the vertex. But in this problem, the vertex value is positive at $ f(1) = 3 $.

Q: Can I use the discriminant to solve this?

Yes! The discriminant \( b^2 - 4ac = 16 - 40 = -24 tells us there are no real roots. Combined with the upward opening, this confirms the function stays positive.

Q: What does 'no solution' mean for inequalities?

'No solution' means there are no x-values that make the inequality true. In this case, \( f(x) is never satisfied because the function is always positive.

Quadratic Inequalities with No Solution

Look at the following function:

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

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:

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

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

$f(x) < 0$

Step-by-step solution

To find the values of $x$ where the function $y = 2x^2 - 4x + 5$ is negative:

Step 1: Identify the direction of the parabola:
Since $a = 2$ is positive, the parabola opens upwards, indicating that any potential minimum will be at the vertex.
Step 2: Find the vertex:
The vertex is at $x = -\frac{b}{2a} = -\frac{-4}{2 \times 2} = 1$ .
Substituting $x = 1$ back into the function gives: $f(1) = 2(1)^2 - 4(1) + 5 = 3$ .
Step 3: Determine if the function can be negative:
Since the vertex provides the minimum value of the parabola and this value is positive $f(1) = 3 > 0$ , the function does not have any $x$ values for which $f(x) < 0$ .

Thus, the correct answer is that the function has no negative values.

Final Answer

The function has no negative values.

Key Points to Remember

Essential concepts to master this topic

Parabola Direction: Positive leading coefficient means upward opening parabola
Vertex Formula: Find minimum at $x = -\frac{b}{2a} = 1$
Check: Substitute vertex: $f(1) = 3 > 0$ confirms no negative values ✓

Common Mistakes

Avoid these frequent errors

Setting the quadratic equal to zero
Don't solve $2x^2 - 4x + 5 = 0$ = finding where it equals zero! This only finds roots, not where the function is negative. Always find the vertex first to determine if the parabola dips below the x-axis.

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

FAQ

Everything you need to know about this question

Why can't I just solve the equation by setting it equal to zero?

Setting the quadratic equal to zero finds roots (where it crosses the x-axis). But we need to know where it's negative, which requires checking the vertex and parabola direction first!

How do I know if a parabola opens upward or downward?

Look at the leading coefficient (the number in front of $x^2$ ). If it's positive, the parabola opens upward. If negative, it opens downward.

What if the vertex is negative but the parabola opens upward?

Then the function would have negative values! The parabola would dip below the x-axis around the vertex. But in this problem, the vertex value is positive at $f(1) = 3$ .

Can I use the discriminant to solve this?

Yes! The discriminant $b^2 - 4ac = 16 - 40 = -24 < 0$ tells us there are no real roots. Combined with the upward opening, this confirms the function stays positive.

What does 'no solution' mean for inequalities?

'No solution' means there are no x-values that make the inequality true. In this case, $f(x) < 0$ is never satisfied because the function is always positive.

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Solve the Quadratic: Discover X in y=2x^2-4x+5 Where f(x) < 0

Quadratic Inequalities with No Solution

❤️ 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

Why can't I just solve the equation by setting it equal to zero?

How do I know if a parabola opens upward or downward?

What if the vertex is negative but the parabola opens upward?

Can I use the discriminant to solve this?

What does 'no solution' mean for inequalities?

🌟 Unlock Your Math Potential

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