Solving x²-6x+8 > 0: Analyzing Quadratic Inequality with Graphs

Q: Why do I need to factor the quadratic first?

Factoring $ x^2 - 6x + 8 = (x-2)(x-4) $ makes it easy to find the roots where the function equals zero. These roots are the boundary points that divide the number line into intervals where the function has consistent sign.

Q: How do I know which intervals to test?

The roots create three intervals: before the first root, between the roots, and after the second root. For this problem: $ x , \( 2 , and \( x > 4 $.

Q: What if I get confused about which sign goes with which interval?

Always test specific values! Pick any number in each interval and substitute it into the original function. If the result is positive, that entire interval satisfies $ f(x) > 0 $.

Q: Why isn't the answer just 2 < x < 4?

That's where $ f(x) (negative)! We want \( f(x) > 0 $ (positive). The parabola opens upward, so it's positive outside the roots and negative between them.

Q: How does the graph help me check my answer?

Look where the parabola is above the x-axis (positive values). You can see it's above the axis for $ x and \( x > 4 $, confirming our algebraic solution!

Quadratic Inequalities with Sign Analysis

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is

$f(x)>0$ true?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 For which values is the function positive?

00:03 We want to find the intersection points with the X-axis

00:07 We'll use the quadratic formula

00:13 We'll find the intersection points with the X-axis

00:24 We'll find the positive domains of the function, according to the graph

00:31 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is

$f(x)>0$ true?

Step-by-step solution

To solve the inequality $f(x) > 0$ , we first need to find the roots of the equation $f(x) = x^2 - 6x + 8$ .

1. Find the roots of the quadratic equation:
The quadratic is $x^2 - 6x + 8$ . This can be factored into:
$f(x) = (x - 2)(x - 4) = 0$ .

2. Calculate the roots:
Setting each factor equal to zero gives the roots $x = 2$ and $x = 4$ .

3. Determine the intervals defined by these roots:
The roots divide the x-axis into three intervals: $(-\infty, 2)$ , $(2, 4)$ , and $(4, \infty)$ .

4. Test points in each interval to decide positivity:
- For $x < 2$ , select $x = 1$ : $f(1) = 1^2 - 6(1) + 8 = 3 > 0$ . Thus, $f(x) > 0$ in $(-\infty, 2)$ .
- For $2 < x < 4$ , select $x = 3$ : $f(3) = 3^2 - 6(3) + 8 = -1 < 0$ . Thus, $f(x) < 0$ in $(2, 4)$ .
- For $x > 4$ , select $x = 5$ : $f(5) = 5^2 - 6(5) + 8 = 3 > 0$ . Thus, $f(x) > 0$ in $(4, \infty)$ .

Therefore, the solution to $f(x) > 0$ is when $x < 2$ or $x > 4$ .

The final solution is: $x < 2, 4 < x$ .

Final Answer

$x < 2, 4 < x$

Key Points to Remember

Essential concepts to master this topic

Roots: Factor to find where $f(x) = 0$ divides intervals
Test Points: Check sign in each interval: $f(1) = 3 > 0$ , $f(3) = -1 < 0$
Verify: Graph confirms parabola above x-axis when $x < 2$ or $x > 4$ ✓

Common Mistakes

Avoid these frequent errors

Only solving f(x) = 0 and forgetting to test intervals
Don't just find roots x = 2 and x = 4 and stop = incomplete solution! Finding roots only shows where the function changes sign, not which intervals are positive. Always test a point in each interval to determine where f(x) > 0.

Practice Quiz

Test your knowledge with interactive questions

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x)<0 $$ true?

FAQ

Everything you need to know about this question

Why do I need to factor the quadratic first?

Factoring $x^2 - 6x + 8 = (x-2)(x-4)$ makes it easy to find the roots where the function equals zero. These roots are the boundary points that divide the number line into intervals where the function has consistent sign.

How do I know which intervals to test?

The roots create three intervals: before the first root, between the roots, and after the second root. For this problem: $x < 2$ , $2 < x < 4$ , and $x > 4$ .

What if I get confused about which sign goes with which interval?

Always test specific values! Pick any number in each interval and substitute it into the original function. If the result is positive, that entire interval satisfies $f(x) > 0$ .

Why isn't the answer just 2 < x < 4?

That's where $f(x) < 0$ (negative)! We want $f(x) > 0$ (positive). The parabola opens upward, so it's positive outside the roots and negative between them.

How does the graph help me check my answer?

Look where the parabola is above the x-axis (positive values). You can see it's above the axis for $x < 2$ and $x > 4$ , confirming our algebraic solution!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Equations and Systems of Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime