91Ó°ÊÓ

Understanding the concept of a matrix determinant is crucial when solving certain algebraic problems. In this case, we're looking at a 2x2 matrix where the determinant can be found using the formula \( ad - bc \). The variables \(a\), \(b\), \(c\), and \(d\) correspond to the entries in the matrix, positioned as follows: \[\left|\begin{array}{cc}a & b \c & d\end{array}\right|\].

For our exercise, the matrix is: \[\left|\begin{array}{rr}x & 4 \-1 & x\end{array}\right|\] and its determinant is calculated as \(x \cdot x - (-1) \cdot 4\), simplifying to \(x^2 + 4\). This calculation is essential as it transforms the problem into a familiar form – a quadratic equation.

When faced with a quadratic equation, such as \(x^2 = 16\) derived from our determinant calculation, the quadratic formula is an invaluable tool. This formula is generally stated as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation in the form \(ax^2 + bx + c = 0\).

In the context of our exercise, we can rearrange \(x^2 + 4 = 20\) into \(x^2 - 16 = 0\), making \(a=1\), \(b=0\), and \(c=-16\). Plugging these values into the quadratic formula, we find that \(x = \pm \sqrt{16}\), which simplifies to the two solutions \(x = 4\) and \(x = -4\).

Absolute value equations involve an expression within absolute value bars, such as \( |x| = a \) which indicates that the distance between \(x\) and 0 on a number line is \(a\), regardless of direction. This gives us two possible scenarios: \(x = a\) or \(x = -a\).

In our exercise, while the absolute value is not explicitly present, the determinant's comparison to a concrete number (20) led to a quadratic equation with two potential solutions reflecting the essence of how absolute values deal with two possible 'paths' or signs. Hence, following through the steps of solving the quadratic equation, we reveal the nature of absolute values by finding that \(x\) can be either 4 or -4, both of which are an equal distance from 0 on the number line.