Solve the Quadratic Inequality: x² - 8x + 12 < 0

Q: Why do I need to factor first instead of just solving directly?

Factoring reveals the roots where the expression changes sign! Without factoring $ (x-2)(x-6) $, you can't easily see that the expression switches from positive to negative at x=2 and x=6.

Q: How do I know which intervals to test?

The roots divide the number line into separate regions. For roots at 2 and 6, test one point in each region: $ x , \( 2 , and \( x > 6 $.

Q: What's the difference between < and ≤ in the final answer?

The symbol matters! Since we want $ (strictly less than), we use open intervals \( 2 . If it were \( ≤ 0 $, we'd include the roots: $ 2 ≤ x ≤ 6 $.

Q: Can I use the quadratic formula instead of factoring?

The quadratic formula finds the roots, but you still need to test intervals! Factoring is actually easier here because it immediately shows the roots and makes testing simpler.

Q: What if my test point gives me zero?

If your test point gives zero, you accidentally picked a root! Choose a different point within that interval. Remember, roots are the boundary points, not inside the intervals.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Solve the following equation:

$x^2-8x+12<0$

2

Step-by-step solution

To solve the quadratic inequality $x^2 - 8x + 12 < 0$ , we follow these steps:

Step 1: Factor the quadratic expression. The expression $x^2 - 8x + 12$ can be factored as $(x - 2)(x - 6)$ .
Step 2: Find the roots of the resulting equation. Set $(x - 2)(x - 6) = 0$ to find the roots. Solving gives $x = 2$ and $x = 6$ .
Step 3: Determine the intervals for testing. The roots divide the number line into three intervals: $x < 2$ , $2 < x < 6$ , and $x > 6$ .
Step 4: Test each interval to find where the inequality holds.

For $x < 2$ , choose a test point such as $x = 0$ . Then $(0 - 2)(0 - 6) = ( -2)( -6) = 12$ , which is greater than 0.
For $2 < x < 6$ , choose a test point such as $x = 4$ . Then $(4 - 2)(4 - 6) = (2)(-2) = -4$ , which is less than 0.
For $x > 6$ , choose a test point such as $x = 7$ . Then $(7 - 2)(7 - 6) = (5)(1) = 5$ , which is greater than 0.

Step 5: Conclude the solution. The expression $(x - 2)(x - 6)$ is negative in the interval $2 < x < 6$ .

Therefore, the solution to the inequality is $2 < x < 6$ .

3

Final Answer

$2 < x < 6$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Convert $x^2 - 8x + 12 < 0$ to $(x-2)(x-6) < 0$
Test Points: Choose values in each interval: x=0 gives 12>0, x=4 gives -4<0
Verify Solution: Check endpoints excluded: at x=2 and x=6, expression equals 0 ✓

Common Mistakes

Avoid these frequent errors

Including the roots in the solution set
Don't write $2 ≤ x ≤ 6$ for strict inequality = includes points where expression equals zero! The original inequality is strictly less than zero, not less than or equal to. Always use open intervals $2 < x < 6$ for strict inequalities.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why do I need to factor first instead of just solving directly?

+

Factoring reveals the roots where the expression changes sign! Without factoring $(x-2)(x-6)$ , you can't easily see that the expression switches from positive to negative at x=2 and x=6.

How do I know which intervals to test?

+

The roots divide the number line into separate regions. For roots at 2 and 6, test one point in each region: $x < 2$ , $2 < x < 6$ , and $x > 6$ .

What's the difference between < and ≤ in the final answer?

+

The symbol matters! Since we want $< 0$ (strictly less than), we use open intervals $2 < x < 6$ . If it were $≤ 0$ , we'd include the roots: $2 ≤ x ≤ 6$ .

Can I use the quadratic formula instead of factoring?

+

The quadratic formula finds the roots, but you still need to test intervals! Factoring is actually easier here because it immediately shows the roots and makes testing simpler.

What if my test point gives me zero?

+

If your test point gives zero, you accidentally picked a root! Choose a different point within that interval. Remember, roots are the boundary points, not inside the intervals.

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Solve the Quadratic Inequality: x² - 8x + 12 < 0

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