Solve y=-(4x+32)²: Finding Values Where Function is Negative

Q: Why is the function negative for almost all x-values?

Because $ (4x + 32)^2 $ is always positive or zero, and the negative sign in front makes it negative or zero. It's only zero when $ x = -8 $.

Q: How do I find when the squared expression equals zero?

Set the expression inside the parentheses equal to zero: $ 4x + 32 = 0 $. Then solve: $ 4x = -32 $, so $ x = -8 $.

Q: What's the difference between f(x) < 0 and f(x) ≤ 0?

f(x) means strictly less than zero (excludes where f(x) = 0). f(x) ≤ 0 includes where the function equals zero. Here, we want strictly negative, so we exclude $ x = -8 $.

Q: Can a squared expression ever be negative?

No! Any real number squared is always positive or zero. But when you put a negative sign in front of the squared expression, that can make the result negative.

Q: Why do we write the answer as x ≠ -8?

This notation means "x can be any real number except -8". At every other value of x, the function is negative, which is exactly what f(x)

Quadratic Functions with Negative Leading Coefficient

Look at the function below:

$y=-\left(4x+32\right)^2$

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:

$y=-\left(4x+32\right)^2$

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

$f(x) < 0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Analyze the structure of the function.
Step 2: Determine when the squared term is zero.
Step 3: Apply the negative sign.
Step 4: Conclude when the function is negative.

Now, let's work through each step:

Step 1: Analyze the function $y = -\left(4x + 32\right)^2$ . Note that the expression $(4x + 32)$ is squared, and any square of a real number is non-negative.

Step 2: For $(4x + 32)^2$ to equal zero, solve $4x + 32 = 0$ .

Solve $4x + 32 = 0$ :

$4x = -32 \quad \Rightarrow \quad x = -8$

Step 3: Apply the negative sign: Since $(4x + 32)^2$ can only be zero or positive, applying the negative sign results in:

$-\left(4x + 32\right)^2 \leq 0$

Step 4: Conclude that $f(x) < 0$ for all values except when $(4x + 32)^2$ is zero. Thus, the function is negative for all $x$ except $x = -8$ .

Therefore, the solution is $x \ne -8$ .

Final Answer

$x\ne-8$

Key Points to Remember

Essential concepts to master this topic

Structure: Negative squared terms are always non-positive
Technique: Find zeros by setting $4x + 32 = 0$ to get $x = -8$
Check: At $x = -8$ , function equals 0; elsewhere it's negative ✓

Common Mistakes

Avoid these frequent errors

Thinking the function is never negative
Don't assume that because there's a squared term, the function is always positive = ignoring the negative sign! The negative outside makes $-(\text{positive}) = \text{negative}$ . Always consider the negative sign affects the entire squared expression.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why is the function negative for almost all x-values?

Because $(4x + 32)^2$ is always positive or zero, and the negative sign in front makes it negative or zero. It's only zero when $x = -8$ .

How do I find when the squared expression equals zero?

Set the expression inside the parentheses equal to zero: $4x + 32 = 0$ . Then solve: $4x = -32$ , so $x = -8$ .

What's the difference between f(x) < 0 and f(x) ≤ 0?

f(x) < 0 means strictly less than zero (excludes where f(x) = 0). f(x) ≤ 0 includes where the function equals zero. Here, we want strictly negative, so we exclude $x = -8$ .

Can a squared expression ever be negative?

No! Any real number squared is always positive or zero. But when you put a negative sign in front of the squared expression, that can make the result negative.

Why do we write the answer as x ≠ -8?

This notation means "x can be any real number except -8". At every other value of x, the function is negative, which is exactly what f(x) < 0 asks for.

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Solve y=-(4x+32)²: Finding Values Where Function is Negative

Quadratic Functions with Negative Leading Coefficient

❤️ 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 is the function negative for almost all x-values?

How do I find when the squared expression equals zero?

What's the difference between f(x) < 0 and f(x) ≤ 0?

Can a squared expression ever be negative?

Why do we write the answer as x ≠ -8?

🌟 Unlock Your Math Potential

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