Solve (x-4.4)(x-2.3) > 0: Finding Positive Values of a Quadratic Function

Q: Why does the product change from positive to negative at the roots?

At each root, one factor changes from negative to positive (or vice versa). Since we multiply the factors, the overall sign of the product flips at each zero point.

Q: How do I remember which intervals are positive?

Test a point! Pick any number in each interval and substitute it. For example, try x = 0: $(0-4.4)(0-2.3) = (-4.4)(-2.3) = 10.12 > 0$

Q: What's the difference between > 0 and ≥ 0?

With > 0, we exclude the roots (2.3 and 4.4) because the expression equals zero there, not greater than zero. Use ≥ 0 only if you want to include where it equals zero.

Q: Can I expand the expression first instead?

You could expand to get $x^2 - 6.7x + 10.12 > 0$, but the factored form is easier! You can immediately see the roots and test intervals without using the quadratic formula.

Q: Why are there two separate solution regions?

The parabola opens upward (positive leading coefficient), so it's positive on the outside of the roots and negative between them. This creates two separate regions where the inequality is true.

Quadratic Inequalities with Factored Form

Look at the function below:

$y=\left(x-4.4\right)\left(x-2.3\right)$

Then 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 function below:

$y=\left(x-4.4\right)\left(x-2.3\right)$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To solve this problem, we need to analyze where the expression $(x - 4.4)(x - 2.3)$ is greater than zero. We have two roots, $x = 4.4$ and $x = 2.3$ , which divide the number line into three intervals: $x < 2.3$ , $2.3 < x < 4.4$ , and $x > 4.4$ .

Let's check these intervals:

For $x < 2.3$ , both $(x - 4.4)$ and $(x - 2.3)$ are negative, resulting in a positive product.
For $2.3 < x < 4.4$ , $(x - 4.4)$ is negative and $(x - 2.3)$ is positive, resulting in a negative product.
For $x > 4.4$ , both $(x - 4.4)$ and $(x - 2.3)$ are positive, resulting in a positive product.

Thus, the expression $(x - 4.4)(x - 2.3) > 0$ holds in the intervals $x < 2.3$ and $x > 4.4$ .

Therefore, the solution is $x > 4.4$ or $x < 2.3$ .

Final Answer

$x > 4.4$ or $x < 2.3$

Key Points to Remember

Essential concepts to master this topic

Rule: Find zeros first, then test intervals between roots
Technique: At x = 0: (0-4.4)(0-2.3) = 10.12 > 0 positive
Check: Verify boundaries: at x = 2.3 and x = 4.4, expression equals zero ✓

Common Mistakes

Avoid these frequent errors

Incorrectly determining sign in each interval
Don't just guess positive or negative in each region = wrong solution set! Students often mix up which intervals are positive. Always test a specific value from each interval by substituting into the original expression.

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

Why does the product change from positive to negative at the roots?

At each root, one factor changes from negative to positive (or vice versa). Since we multiply the factors, the overall sign of the product flips at each zero point.

How do I remember which intervals are positive?

Test a point! Pick any number in each interval and substitute it. For example, try x = 0: $(0-4.4)(0-2.3) = (-4.4)(-2.3) = 10.12 > 0$

What's the difference between > 0 and ≥ 0?

With > 0, we exclude the roots (2.3 and 4.4) because the expression equals zero there, not greater than zero. Use ≥ 0 only if you want to include where it equals zero.

Can I expand the expression first instead?

You could expand to get $x^2 - 6.7x + 10.12 > 0$ , but the factored form is easier! You can immediately see the roots and test intervals without using the quadratic formula.

Why are there two separate solution regions?

The parabola opens upward (positive leading coefficient), so it's positive on the outside of the roots and negative between them. This creates two separate regions where the inequality is true.

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Solve (x-4.4)(x-2.3) > 0: Finding Positive Values of a Quadratic Function

Quadratic Inequalities with Factored Form

❤️ 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 does the product change from positive to negative at the roots?

How do I remember which intervals are positive?

What's the difference between > 0 and ≥ 0?

Can I expand the expression first instead?

Why are there two separate solution regions?

🌟 Unlock Your Math Potential

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