Solving the Absolute Value Quadratic: |x² - 6x + 8| = 0

Q: Why don't I need to consider both positive and negative cases here?

Great question! When an absolute value equals zero, there's only one possibility: the expression inside must equal zero. Unlike $ |x| = 5 $ where x could be 5 or -5, $ |x| = 0 $ only happens when x = 0.

Q: How do I factor x² - 6x + 8 quickly?

Look for two numbers that multiply to 8 and add to -6. Since 8 is positive and the middle term is negative, both factors are negative: -2 and -4. So $ x^2 - 6x + 8 = (x-2)(x-4) $.

Q: What if the quadratic doesn't factor nicely?

If factoring is difficult, use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $. The solutions will still be the same, just found through a different method!

Q: Can an absolute value equation have no solutions?

Yes! If the equation were $ |x^2 - 6x + 8| = -3 $, there would be no solutions because absolute values are never negative. But since our equation equals zero, solutions definitely exist.

Q: How can I check my factoring is correct?

Expand your factors using FOIL: $ (x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8 $. If you get back the original expression, your factoring is correct!

Absolute Value Equations with Quadratic Expressions

$|x^2-6x+8|=0$

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Understand the problem

$|x^2-6x+8|=0$

Step-by-step solution

To solve the equation $|x^2 - 6x + 8| = 0$ , recognize that the absolute value of a number is zero if and only if the number itself is zero. Therefore, we set the expression inside the absolute value to zero:

$x^2 - 6x + 8 = 0$

Next, we attempt to factor the quadratic expression:

The expression $x^2 - 6x + 8$ can be factored into:

$(x - 2)(x - 4) = 0$

Now, apply the zero product property, which states if a product equals zero, at least one of the factors must be zero. So, set each factor equal to zero:

$x - 2 = 0$ gives $x = 2$
$x - 4 = 0$ gives $x = 4$

Thus, the solutions to the equation $|x^2 - 6x + 8| = 0$ are:

$x = 2$ and $x = 4$ .

Final Answer

$x=2$ , $x=4$

Key Points to Remember

Essential concepts to master this topic

Key Rule: Absolute value equals zero only when expression inside equals zero
Technique: Factor $x^2 - 6x + 8 = (x-2)(x-4)$ then set each factor to zero
Check: Verify both solutions: $|4-12+8| = |0| = 0$ and $|16-24+8| = |0| = 0$ ✓

Common Mistakes

Avoid these frequent errors

Trying to split absolute value into positive and negative cases
Don't set up $x^2 - 6x + 8 = 0$ and $x^2 - 6x + 8 = -0$ = same equation twice! Since the right side is zero, splitting into cases is unnecessary and wastes time. Always recognize that $|A| = 0$ means $A = 0$ directly.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why don't I need to consider both positive and negative cases here?

Great question! When an absolute value equals zero, there's only one possibility: the expression inside must equal zero. Unlike $|x| = 5$ where x could be 5 or -5, $|x| = 0$ only happens when x = 0.

How do I factor x² - 6x + 8 quickly?

Look for two numbers that multiply to 8 and add to -6. Since 8 is positive and the middle term is negative, both factors are negative: -2 and -4. So $x^2 - 6x + 8 = (x-2)(x-4)$ .

What if the quadratic doesn't factor nicely?

If factoring is difficult, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . The solutions will still be the same, just found through a different method!

Can an absolute value equation have no solutions?

Yes! If the equation were $|x^2 - 6x + 8| = -3$ , there would be no solutions because absolute values are never negative. But since our equation equals zero, solutions definitely exist.

How can I check my factoring is correct?

Expand your factors using FOIL: $(x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8$ . If you get back the original expression, your factoring is correct!

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