Solve (2x-16)² < 0: Finding Values of x in a Quadratic Inequality

Q: Why can't a squared expression be negative?

When you square any real number, the result is always positive or zero. For example: $ 3^2 = 9 $ and $ (-3)^2 = 9 $. Even $ 0^2 = 0 $. There's no real number that becomes negative when squared!

Q: What does it mean when there are 'no values' of x?

It means the inequality has no solution. The set of values that make \( (2x-16)^2 true is empty because this condition is impossible for real numbers.

Q: When does (2x-16)² equal zero?

The expression equals zero when $ 2x - 16 = 0 $, which gives us $ x = 8 $. At this point, $ (2x-16)^2 = 0 $, but it's still not negative.

Q: How is this different from linear inequalities?

Linear inequalities like \( 2x - 16 can have solutions because the expression isn't squared. But when you square an expression, you eliminate the possibility of negative values.

Q: What if the question asked for (2x-16)² > 0?

Then the answer would be all real numbers except x = 8. The squared expression is positive everywhere except at $ x = 8 $ where it equals zero.

Quadratic Inequalities with Squared Expressions

Look at the function below:

$y=\left(2x-16\right)^2$

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:

$y=\left(2x-16\right)^2$

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

$f(x) < 0$

Step-by-step solution

To solve this problem, we must determine when the expression $(2x-16)^2$ is less than zero.

First, consider the expression $(2x-16)^2$ .

Recognize that squaring any real number results in a non-negative number. Thus, $(2x-16)^2$ is always non-negative.
Specifically, for any expression squared, $(a)^2 \geq 0$ for all real numbers $a$ .
Therefore, $(2x-16)^2$ cannot be less than zero for any value of $x$ .

Since a square of any real function is always zero or positive, there are no real values of $x$ for which $(2x-16)^2$ is negative.

Therefore, the conclusion is that there are no values of $x$ that make $(2x-16)^2 < 0$ .

The correct answer is: No $x$ .

Final Answer

No $x$

Key Points to Remember

Essential concepts to master this topic

Rule: Any real number squared is always greater than or equal to zero
Technique: Recognize $(2x-16)^2 \geq 0$ for all values of x
Check: Test any value: $(2(0)-16)^2 = 256 \geq 0$ ✓

Common Mistakes

Avoid these frequent errors

Trying to solve by setting the expression equal to zero
Don't solve $(2x-16)^2 = 0$ to find when it's negative = wrong approach! This finds when the expression equals zero (x = 8), not when it's less than zero. Always remember that squared expressions are never negative.

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 a squared expression be negative?

When you square any real number, the result is always positive or zero. For example: $3^2 = 9$ and $(-3)^2 = 9$ . Even $0^2 = 0$ . There's no real number that becomes negative when squared!

What does it mean when there are 'no values' of x?

It means the inequality has no solution. The set of values that make $(2x-16)^2 < 0$ true is empty because this condition is impossible for real numbers.

When does (2x-16)² equal zero?

The expression equals zero when $2x - 16 = 0$ , which gives us $x = 8$ . At this point, $(2x-16)^2 = 0$ , but it's still not negative.

How is this different from linear inequalities?

Linear inequalities like $2x - 16 < 0$ can have solutions because the expression isn't squared. But when you square an expression, you eliminate the possibility of negative values.

What if the question asked for (2x-16)² > 0?

Then the answer would be all real numbers except x = 8. The squared expression is positive everywhere except at $x = 8$ where it equals zero.

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Solve (2x-16)² < 0: Finding Values of x in a Quadratic Inequality

Quadratic Inequalities with Squared Expressions

❤️ 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 a squared expression be negative?

What does it mean when there are 'no values' of x?

When does (2x-16)² equal zero?

How is this different from linear inequalities?

What if the question asked for (2x-16)² > 0?

🌟 Unlock Your Math Potential

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