Solve: When is -(2x-2¼)² Less Than Zero? Complete Solution Guide

Q: Why does the negative sign make the function negative almost everywhere?

The expression $ \left(2x - 2\frac{1}{4}\right)^2 $ is always positive or zero because it's a square. When you put a negative sign in front, you get negative or zero values!

Q: How do I find where the function equals zero?

Set the expression inside the square equal to zero: $ 2x - 2\frac{1}{4} = 0 $. This gives you $ x = 1\frac{1}{8} $, which is the only point where f(x) = 0.

Q: What does 'x ≠ 1⅛' actually mean?

It means x can be any real number except $ 1\frac{1}{8} $. So x could be 0, 5, -10, 1.124, 1.126, etc. - just not exactly $ 1\frac{1}{8} $!

Q: How can I convert between 1⅛ and decimal form?

$ 1\frac{1}{8} = 1 + \frac{1}{8} = 1 + 0.125 = 1.125 $. Remember that $ \frac{1}{8} = 0.125 $ because 1 ÷ 8 = 0.125.

Q: Why isn't the answer 'x 1⅛'?

Those would be correct for a regular parabola, but this one opens downward! The function is negative on both sides of the vertex, so we exclude only the vertex point itself.

Quadratic Inequalities with Negative Leading Coefficient

Look at the function below:

$y=-\left(2x-2\frac{1}{4}\right)^2$

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:

$y=-\left(2x-2\frac{1}{4}\right)^2$

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

$f(x) < 0$

Step-by-step solution

The function is given in vertex form: $y = -\left(2x - 2\frac{1}{4}\right)^2$ , which translates to $y = -\left(2x - 2.25\right)^2$ . The vertex occurs when the expression inside the square is zero, which is at $x = 1\frac{1}{8}$ . This is the maximum point due to the negative coefficient, making the function value at the vertex equal to zero.

For $f(x) < 0$ , the square term $\left(2x - 2.25\right)^2$ must be non-zero. Thus, set $2x - 2\frac{1}{4} = 0$ to find the $x$ that needs to be excluded:

$2x - 2.25 = 0$

$2x = 2.25$

$x = 1.125$ or $x = 1\frac{1}{8}$

Therefore, for $f(x) < 0$ , $x$ should not be equal to $1\frac{1}{8}$ .

The correct condition is: $x \neq 1\frac{1}{8}$ .

Final Answer

$x\ne1\frac{1}{8}$

Key Points to Remember

Essential concepts to master this topic

Vertex Form Recognition: Negative coefficient makes parabola open downward with maximum
Critical Value: Set $2x - 2\frac{1}{4} = 0$ gives $x = 1\frac{1}{8}$
Check: At vertex f(x) = 0, everywhere else f(x) < 0 ✓

Common Mistakes

Avoid these frequent errors

Thinking the function is never negative
Don't assume f(x) < 0 has no solutions because there's a negative sign = missing that it's only zero at one point! The negative coefficient means the parabola opens downward, making f(x) negative everywhere except the vertex. Always remember that -(positive number) is negative.

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 does the negative sign make the function negative almost everywhere?

The expression $\left(2x - 2\frac{1}{4}\right)^2$ is always positive or zero because it's a square. When you put a negative sign in front, you get negative or zero values!

How do I find where the function equals zero?

Set the expression inside the square equal to zero: $2x - 2\frac{1}{4} = 0$ . This gives you $x = 1\frac{1}{8}$ , which is the only point where f(x) = 0.

What does 'x ≠ 1⅛' actually mean?

It means x can be any real number except $1\frac{1}{8}$ . So x could be 0, 5, -10, 1.124, 1.126, etc. - just not exactly $1\frac{1}{8}$ !

How can I convert between 1⅛ and decimal form?

$1\frac{1}{8} = 1 + \frac{1}{8} = 1 + 0.125 = 1.125$ . Remember that $\frac{1}{8} = 0.125$ because 1 ÷ 8 = 0.125.

Why isn't the answer 'x < 1⅛' or 'x > 1⅛'?

Those would be correct for a regular parabola, but this one opens downward! The function is negative on both sides of the vertex, so we exclude only the vertex point itself.

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Solve: When is -(2x-2¼)² Less Than Zero? Complete Solution Guide

Quadratic Inequalities 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 does the negative sign make the function negative almost everywhere?

How do I find where the function equals zero?

What does 'x ≠ 1⅛' actually mean?

How can I convert between 1⅛ and decimal form?

Why isn't the answer 'x < 1⅛' or 'x > 1⅛'?

🌟 Unlock Your Math Potential

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