Solve (3x+30)² > 0: Finding Positive Values of a Quadratic Function

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

Because when x = -10, we get $ (3(-10) + 30)^2 = 0^2 = 0 $, and 0 is not greater than 0. We need strictly positive values!

Q: How do I remember when squared expressions equal zero?

A squared expression equals zero only when the base equals zero. So solve $ 3x + 30 = 0 $ to find the excluded value: x = -10.

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

The symbol > 0 means 'strictly greater than zero' (excludes 0), while ≥ 0 means 'greater than or equal to zero' (includes 0). This problem asks for > 0!

Q: Can I just say x > -10 instead of x ≠ -10?

No! The function is positive for both x > -10 and x $ x \neq -10 $ to include all positive values.

Q: How can I verify my answer is correct?

Test values on both sides of x = -10: try x = -11 and x = -9. Both should give positive results: $ (3(-11) + 30)^2 = (-3)^2 = 9 > 0 $ ✓

Squared Functions with Zero Analysis

Look at the function below:

$y=\left(3x+30\right)^2$

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:

$y=\left(3x+30\right)^2$

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

$f(x) > 0$

Step-by-step solution

To determine the values of $x$ for which the function $y = (3x + 30)^2$ is greater than zero, consider the following steps:

Step 1: Recognize the structure of the function. The function is of the form $(\text{expression})^2$ . For the function to be greater than zero, the expression inside the square must not equal zero.
Step 2: Solve $3x + 30 = 0$ to find when the function equals zero.
Subtract 30 from both sides: $3x = -30$ .
Divide by 3: $x = -10$ .
Step 3: Exclude $x = -10$ from the domain where the function is greater than 0. For all $x \neq -10$ , $(3x + 30)^2$ is positive because it results from squaring a non-zero real number.

Therefore, $f(x) > 0$ for all $x \neq -10$ .

The correct answer is $x\ne-10$ .

Final Answer

$x\ne-10$

Key Points to Remember

Essential concepts to master this topic

Rule: Squared expressions are positive except when the base equals zero
Technique: Set 3x + 30 = 0, solve to find x = -10
Check: Test x ≠ -10: (3(-9) + 30)² = 3² = 9 > 0 ✓

Common Mistakes

Avoid these frequent errors

Thinking squared functions are always positive
Don't assume (3x + 30)² > 0 for ALL x values = includes zero incorrectly! When 3x + 30 = 0, the squared result is 0, not positive. Always find where the expression inside equals zero and exclude that value.

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 when x = -10, we get $(3(-10) + 30)^2 = 0^2 = 0$ , and 0 is not greater than 0. We need strictly positive values!

How do I remember when squared expressions equal zero?

A squared expression equals zero only when the base equals zero. So solve $3x + 30 = 0$ to find the excluded value: x = -10.

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

The symbol > 0 means 'strictly greater than zero' (excludes 0), while ≥ 0 means 'greater than or equal to zero' (includes 0). This problem asks for > 0!

Can I just say x > -10 instead of x ≠ -10?

No! The function is positive for both x > -10 and x < -10. Only x = -10 makes it zero. So we need $x \neq -10$ to include all positive values.

How can I verify my answer is correct?

Test values on both sides of x = -10: try x = -11 and x = -9. Both should give positive results: $(3(-11) + 30)^2 = (-3)^2 = 9 > 0$ ✓

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Solve (3x+30)² > 0: Finding Positive Values of a Quadratic Function

Squared Functions with Zero Analysis

❤️ 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 remember when squared expressions equal zero?

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

Can I just say x > -10 instead of x ≠ -10?

How can I verify my answer is correct?

🌟 Unlock Your Math Potential

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