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Chapter 2: Problem 35

Explain what is unusual about the solution set of the inequality. $$4 x^{2}-4 x+1 \leq 0$$

Short Answer

Expert verified
The solution of the inequality \(4 x^{2}-4 x+1 \leq 0\) is \(x = 0.5\). The unusual part about the solution is the realization that the inequality will only hold for one specific value of \(x\), that's because a square of a number (which is always non-negative) is being compared to 0.

Step by step solution

01

Rewrite the inequality

Rewrite the given inequality \(4x^{2}-4x+1 \leq 0\) in term of perfect square. So, \(4x^{2}-4x+1 = (2x-1)^{2} \leq 0\)
02

Understanding a square property

We know that the square of any real number is always greater or equal to zero, and thus \((2x-1)^2\) is always non-negative.
03

Solve the inequality.

The equality \((2x-1)^2 = 0\) will only hold for \(x = 0.5\). Since it can't be less than 0, so solution set of the inequality \(4x^{2}-4x+1 \leq 0\) is \(x = 0.5\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomials
In the realm of quadratic inequalities, perfect square trinomials are standout expressions. A perfect square trinomial is a type of quadratic expression that can be factored into a binomial squared. For instance, in the original exercise, the quadratic expression \(4x^{2} - 4x + 1\) can be rewritten as \((2x-1)^{2}\). This step significantly simplifies solving the quadratic inequality, showcasing the elegance of recognizing patterns in mathematics.
  • A perfect square trinomial takes the form \(a^2 + 2ab + b^2\).
  • When factored, it results in \((a+b)^2\).
  • In our example, \(a = 2x\) and \(b = -1\).
Recognizing and transforming a quadratic equation into a perfect square trinomial simplifies determining the possible solutions as it brings us closer to understanding the behavior of the inequality.
Square Property
The square property is a crucial concept when dealing with squared terms in mathematics. The principle focuses on the fact that the square of any real number is always non-negative. This means, for any expression like \((2x-1)^2\), the value will either be zero or positive.
  • Non-negativity plays a role in determining inequalities.
  • Since \( (2x-1)^2 \geq 0\), it cannot be less than zero.
These properties are vital, especially when solving inequalities. It means any solution must fulfill the condition where the squared term does not yield negative results. Understanding this narrows down potential solutions and focuses our attention on specific values.
Solution Set Analysis
Solution set analysis involves determining the precise set of values that satisfy an inequality or equation. In the context of \((2x-1)^2 \leq 0\), we noticed that the expression cannot be less than zero due to the square property. The equality holds only when \((2x-1)^2 = 0\).
  • In any equation of the form \(a^2 \leq 0\), the only solution is when \(a = 0\).
  • In our problem, \((2x-1) = 0\) leads directly to the solution \(x = 0.5\).
Conducting a thorough solution set analysis ensures completeness and accuracy in determining the solution to the inequality, revealing that the only solution to the inequality \(4x^{2}-4x+1 \leq 0\) is indeed \(x = 0.5\).

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Most popular questions from this chapter

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. $$\text { Vertex: }\left(-\frac{1}{4}, \frac{3}{2}\right) ; \text { point: }(-2,0)$$

Find the values of \(b\) such that the function has the given maximum or minimum value. $$f(x)=-x^{2}+b x-75 ; \text { Maximum value: } 25$$

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$g(x)=x^{2}-8 x$$

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (-2,-2)\(;\) point: (-1,0)
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