Solve (2x-16)² > 0: Finding Values Where Function is Positive

Q: Why isn't the answer 'true for all values of x'?

Because at $ x = 8 $, the function equals zero, not positive. Since we need $ f(x) > 0 $ (strictly greater than), we must exclude the point where it equals zero.

Q: How do I find where a perfect square equals zero?

Set the expression inside the square equal to zero: $ 2x - 16 = 0 $. Solve to get $ x = 8 $. This is the only point where the square equals zero.

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

≥ 0 means 'greater than or equal to zero' (includes zero), while > 0 means 'strictly greater than zero' (excludes zero). Perfect squares satisfy ≥ 0 always, but > 0 excludes the zero point.

Q: Can I graph this to check my answer?

Yes! The graph of $ y = (2x - 16)^2 $ is a parabola opening upward that touches the x-axis at $ x = 8 $ and is positive everywhere else.

Q: Why does the square make the function always non-negative?

Any real number squared gives a non-negative result. Whether $ 2x - 16 $ is positive or negative, squaring it always gives zero or a positive value.

Quadratic Inequalities with Perfect Squares

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 determine when the function $y = (2x - 16)^2$ is greater than zero, we observe the following:

The function's expression, $(2x - 16)^2$ , is a square and thus always non-negative ( $\geq 0$ ).
The expression will equate to zero when the inside term is zero: $2x - 16 = 0$ .
Solve the equation $2x - 16 = 0$ to find $x = 8$ .
Therefore, $(2x - 16)^2 = 0$ only at $x = 8$ .
For $y > 0$ , $(2x - 16)^2$ must be greater than zero, which occurs for all $x$ except $x = 8$ .

$x \ne 8$

Final Answer

$x\ne8$

Key Points to Remember

Essential concepts to master this topic

Rule: Perfect squares are always non-negative for all real numbers
Technique: Set (2x - 16)² = 0 to find x = 8
Check: Test x = 5: (2(5) - 16)² = 36 > 0 ✓

Common Mistakes

Avoid these frequent errors

Solving as if it equals zero instead of greater than zero
Don't solve (2x - 16)² = 0 and think that's the final answer = x = 8 only! This gives you where the function equals zero, not where it's positive. Always remember that squares are positive everywhere except at the zero point, so the answer is x ≠ 8.

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 isn't the answer 'true for all values of x'?

Because at $x = 8$ , the function equals zero, not positive. Since we need $f(x) > 0$ (strictly greater than), we must exclude the point where it equals zero.

How do I find where a perfect square equals zero?

Set the expression inside the square equal to zero: $2x - 16 = 0$ . Solve to get $x = 8$ . This is the only point where the square equals zero.

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

≥ 0 means 'greater than or equal to zero' (includes zero), while > 0 means 'strictly greater than zero' (excludes zero). Perfect squares satisfy ≥ 0 always, but > 0 excludes the zero point.

Can I graph this to check my answer?

Yes! The graph of $y = (2x - 16)^2$ is a parabola opening upward that touches the x-axis at $x = 8$ and is positive everywhere else.

Why does the square make the function always non-negative?

Any real number squared gives a non-negative result. Whether $2x - 16$ is positive or negative, squaring it always gives zero or a positive value.

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Solve (2x-16)² > 0: Finding Values Where Function is Positive

Quadratic Inequalities with Perfect Squares

❤️ 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 isn't the answer 'true for all values of x'?

How do I find where a perfect square equals zero?

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

Can I graph this to check my answer?

Why does the square make the function always non-negative?

🌟 Unlock Your Math Potential

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