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A quadratic equation is a type of polynomial equation of the second degree. This means it includes a squared term as the highest power of the variable. It follows the general form:

• \textbf{Standard form}: \( ax^2 + bx + c = 0 \)
  
  Where:
  
  • \textbf{a, b, c} are constants
  • \textbf{x} is the variable

For instance, in the equation from the exercise \(0.1 p^{2}-0.4 p + 0.1 = 0\),\,\(a = 0.1\),\,\(b = -0.4\),\,\(c = 0.1\).
This type of equation creates a parabolic graph, which might open upwards or downwards depending on the coefficient 'a'.
Finding the roots of the quadratic equation helps identify where the parabola intersects the x-axis. In the provided problem, the goal is to find values of \(p \) where the equation equals zero; hence the x-intercepts or roots.

Solving quadratic equations is fundamental in algebra. There are various methods, including:

• Factoring
• Using the quadratic formula (\( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \))
• Completing the square

In this exercise, we focus on completing the square. This technique involves several steps:

*Step 1: Simplify the equation*
First, eliminate any decimals to make the math easier. Multiply the entire equation by 10: \( 0.1 \cdot 10 p^{2} - 0.4 \cdot 10 p + 0.1 \cdot 10 =0 \). This yields \( p^{2} - 4 p + 1 =0 \)
*Step 2: Move the constant term*
Rewrite the equation as \( p^2 - 4p = -1 \). This isolates the constant term to facilitate completing the square.
*Step 3: Complete the square*
Here, take Half of the linear coefficient (-4), divide by 2, then square it \(\left(\frac{-4}{2}\right)^2 = 4\). Add this number to both sides to balance the equation: \( p^2 - 4p + 4 = -1 + 4 \).
*Step 4: Write as a perfect square trinomial*
Now, the left side \( p^2 - 4p + 4\) is a perfect square trinomial and can be written as \(\left(p – 2\right)^2\). The equation becomes: \( (p-2)^2 = 3 \)
*Step 5: Solve for \(p\)*
Take the square root of both sides \( \pm\sqrt{(p-2)^2}=\pm\sqrt{3} \), resulting in \( p-2 = \pm\sqrt{3}\). Simplifying further gives \( p = 2 + \sqrt{3} \) or \( p = 2 - \sqrt{3} \)

Following these explicit steps will guide you to the correct roots of the quadratic equation.

Completing the square is a valuable algebraic method to solve quadratic equations. It transforms a quadratic equation into a perfect square trinomial. Here's a concise overview of why the steps work as they do:

• \textbf{Isolates variable terms}: By moving the constant term, we prepare the equation for the next steps.
• \textbf{Balances the equation}: Adding the same value to both sides keeps the equation balanced.
• \textbf{Perfect square trinomial}: A perfect square trinomial factors neatly into \((a+b)^2\), simplifying the solution process.

Completing the square is not only useful for solving quadratics but also essential for other algebraic manipulations like deriving the vertex form of a parabolic equation and solving circle equations.

Here’s a practical tip: Always ensure the coefficient of \(x^2\) (or \(p^2\) in this case) equals 1 before completing the square. If it doesn't, factor it out or adjust the equation proportionally.
Following a systematic approach like the one shown in this exercise yields reliable and accurate solutions. Familiarize yourself with the steps to master this helpful algebraic technique.