Problem 44 - Algebra 2 and Trigonometry

Question

In \(43-46,\) solve each equation for the variable. $$ a^{2}=196 $$

Step-by-Step Solution

Answer

The solutions are \(a = 14\) and \(a = -14\).

Step 1: Recognizing the Equation Type

Notice that the equation given is a quadratic equation because it contains the variable squared, specifically, it is of the format \(a^2 = c\). This type requires us to find the square root to solve for \(a\).

Step 2: Taking the Square Root of Both Sides

To solve the equation \(a^2 = 196\), take the square root of both sides to isolate \(a\). Thus, we write: \(a = \pm \sqrt{196}\).

Step 3: Calculate the Square Root

Calculate the square root of 196. Knowing that \(196\) is a perfect square, we find that \(\sqrt{196} = 14\). Therefore, \(a = \pm 14\).

Step 4: Interpretation of the Solution

The equation \(a^2 = 196\) has two solutions since both \(14^2 = 196\) and \((-14)^2 = 196\). Therefore, \(a = 14\) or \(a = -14\).

Key Concepts

Solution of Equations

Solving equations is a fundamental concept in algebra, where the objective is to find the value of unknown variables that make the equation true. There are different types of equations, and each type has specific methods for solving them.
Quadratic equations are a class of polynomial equations, typically in the form of \( ax^2 + bx + c = 0 \). In this particular exercise, the quadratic equation is presented in its simplest form as \( a^2 = c \). The aim is to determine the value(s) of \( a \) that satisfy the equation.
When solving quadratic equations like this one, an effective strategy is to isolate the variable by employing operations that "undo" the variable's complexity. Since the variable \( a \) is squared, taking the square root allows the equation to be simplified to the first power of the variable. This process will often yield two potential solutions (positive and negative roots) due to the properties of squares. The critical point here is to pay attention to details and carefully execute each step to ensure the correct solutions are obtained.

Square Roots

A square root is a number that, when multiplied by itself, yields the original number under the square root. For instance, the square root of 196 is a pivotal step in solving the quadratic equation \( a^2 = 196 \). Understanding and applying square roots is vital, especially when dealing with quadratic equations.
To solve for \( a \), we take the square root of both sides of the equation. Mathematically, this is expressed as \( a = \pm \sqrt{196} \). The symbol \( \pm \) indicates that there are typically two roots: one positive and one negative. This arises because both positive and negative values for \( a \) will satisfy the original squared equation:

• \( 14 \times 14 = 196 \)
• \( -14 \times -14 = 196 \)

Understanding and calculating square roots accurately is crucial as any errors can lead to incorrect results in the solution process.

Perfect Squares

A perfect square is an integer that is the square of another integer. In solving quadratic equations like \( a^2 = 196 \), recognizing perfect squares can significantly simplify the solution process. Perfect squares are notable because their square roots are whole numbers, making calculations straightforward.
In this exercise, the number 196 is identified as a perfect square. This means there is an integer that squares to give 196. Calculating \( \sqrt{196} \) results in 14, an integer, which simplifies the solution.
This property is advantageous because it removes ambiguity, offering an exact and clear solution. Recognizing perfect squares helps streamline solving quadratic equations, making them less intimidating. It also aids in verifying calculations, especially in academic settings where precision is crucial. Therefore, becoming familiar with perfect squares can boost one's confidence and efficiency in mathematics.