How many more unit tiles need to be added to the expression $x^2+4 x+3$ in order to form a perfect square trinomial?
To determine how many unit tiles need to be added to the expression \(x^2 + 4x + 3\) to form a perfect square trinomial, we first recall the general form of such a trinomial. A perfect square trinomial follows the pattern \((x + k)^2 = x^2 + 2kx + k^2\). Our goal is to find a constant \(n\) such that \(x^2 + 4x + 3 + n\) can be written in this form.
Comparing our given expression \(x^2 + 4x + 3 + n\) to the standard form \(x^2 + 2kx + k^2\), we equate the coefficients of the linear terms:
\(2k = 4\)
Dividing both sides by 2, we find:
\(k = 2\)
Now, substituting \(k = 2\) into the expression for the constant term, \(k^2\), we get:
\(k^2 = 2^2 = 4\)
For our expression \(x^2 + 4x + 3 + n\) to be a perfect square, the constant term must be 4. Therefore:
\(3 + n = 4\)
Solving for \(n\), we get \(n = 4 - 3 = 1\).
By adding 1 unit tile to the expression \(x^2 + 4x + 3\), we obtain \(x^2 + 4x + 4\). This expression is a perfect square trinomial, as it factors into \((x + 2)^2\). Since we added 1 unit tile to achieve this, the correct answer is 1, which corresponds to option (b).
In a real-life construction or DIY project, imagine you are arranging tiles on a rectangular floor. If you have an arrangement that forms a nearly square shape, finding how many additional tiles you need to complete a perfect square is exactly like this problem. For example, if you have an area represented by \(x^2 + 4x + 3\) and you want to perfectly frame it into a square of dimensions \((x+2) \times (x+2)\), you need to calculate the difference to identify how many extra tiles (\(n\)) are required to fill the remaining space. This ensures your layout is perfectly balanced and aesthetic.