Find the zeros of the quadratic polynomial 4x^2 - 4x + 1.
Understand the Problem
The question is asking to find the zeros of the quadratic polynomial given by the expression 4x^2 - 4x + 1. This involves applying the quadratic formula or factoring the polynomial to determine the values of x where the polynomial equals zero.
Answer
The zeros are $x = \frac{1}{2}$.
Answer for screen readers
The zeros of the polynomial $4x^2 - 4x + 1$ are $x = \frac{1}{2}$.
Steps to Solve
- Identify the coefficients In the quadratic polynomial $4x^2 - 4x + 1$, identify the coefficients:
- $a = 4$
- $b = -4$
- $c = 1$
- Apply the quadratic formula Use the quadratic formula to find the zeros of the polynomial. The formula is given by:
$$ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} $$
Substituting in our coefficients:
$$ x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 4 \cdot 1}}}}{2 \cdot 4} $$
- Calculate the discriminant Calculate the discriminant $b^2 - 4ac$:
$$ (-4)^2 - 4 \cdot 4 \cdot 1 = 16 - 16 = 0 $$
- Find the zeros Since the discriminant is 0, there is one real double root. Plugging back into the formula:
$$ x = \frac{{4 \pm 0}}{8} $$
This simplifies to:
$$ x = \frac{4}{8} = \frac{1}{2} $$
- Conclusion The quadratic has a double root at $x = \frac{1}{2}$.
The zeros of the polynomial $4x^2 - 4x + 1$ are $x = \frac{1}{2}$.
More Information
When the discriminant of a quadratic equation is zero, it indicates that the quadratic has exactly one unique solution, also known as a double root. This means the graph of the quadratic touches the x-axis at that point but does not cross it.
Tips
- Forgetting to calculate the discriminant can lead to missing the fact that there might be one or two roots.
- Miscalculating the coefficients can cause errors in using the quadratic formula.
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