๐ Why Two Solutions Exist for Square Roots
When solving equations involving squares, such as $x^2 = k$, where $k$ is a positive number, we often encounter the need to take the square root of both sides. The reason we consider both positive and negative roots stems from the very definition of a square root.
- ๐ Definition of Square Root: The square root of a number $k$ is a value that, when multiplied by itself, equals $k$. Mathematically, if $x^2 = k$, then $x$ is a square root of $k$.
- โ Positive and Negative Roots: Consider $x^2 = 9$. Both $3$ and $-3$ satisfy this equation because $(3)(3) = 9$ and $(-3)(-3) = 9$. The key is that a negative number multiplied by itself results in a positive number.
- ๐ข Mathematical Representation: When solving for $x$ in $x^2 = k$, we write $x = \pm\sqrt{k}$, where $\pm$ denotes 'plus or minus'. This notation explicitly indicates that both the positive and negative square roots are solutions.
- โ ๏ธ Principal Square Root: It's important to note that the radical symbol $\sqrt{ }$ by itself usually refers to the principal or positive square root. For example, $\sqrt{9} = 3$. However, when solving equations, we must consider both roots.
- โ Solving Equations: When solving an equation like $x^2 = 16$, we take the square root of both sides to get $x = \pm\sqrt{16}$, which gives us $x = 4$ and $x = -4$ as two distinct solutions.
- ๐ Graphical Interpretation: Graphically, the equation $y = x^2$ represents a parabola. A horizontal line $y = k$ (where $k > 0$) intersects the parabola at two points, corresponding to the positive and negative square roots of $k$.
- ๐ก Real-World Analogy: Imagine a square with an area of 25 square meters. The side length could be considered as 5 meters. However, in the context of solving the equation $x^2=25$, we must consider both 5 and -5 as potential solutions to the *equation*, although a negative side length doesn't have a physical meaning in this specific geometric scenario.
๐ Real-World Examples
While negative lengths don't make sense in basic geometry, here are situations where considering both roots is crucial:
- โ๏ธ Physics - Projectile Motion: In projectile motion calculations, time ($t$) is often found by solving quadratic equations. Both positive and negative solutions may arise, but only the positive time is physically relevant (time before the launch doesn't usually matter). However, understanding both roots can provide insights into the mathematical model.
- โก Electrical Engineering - AC Circuits: When analyzing AC circuits, quadratic equations can arise when determining impedance or current. Negative solutions may indicate a phase shift or direction change.
- ๐ Finance - Compound Interest: In some compound interest problems, solving for the interest rate can lead to a quadratic equation. Although a negative interest rate is not practically possible, understanding both roots helps to validate the model's limitations.
๐ Conclusion
The existence of two solutions when taking the square root of $k$ arises from the definition of a square root: both the positive and negative values, when squared, result in $k$. While in some real-world contexts, only the positive root makes sense, recognizing both solutions is crucial for solving equations and understanding the underlying mathematical principles.