Problem 2 - Elementary Algebra

Question

Solve the following quadratic equations. \(b^{2}=144\)

Step-by-Step Solution

Answer

The solutions are \( b = 12 \) and \( b = -12 \).

Step 1 - Understand the equation

The equation given is a quadratic equation: \( b^2 = 144 \). This represents a simple quadratic equation in the form of \( x^2 = c \).

Step 2 - Isolate the variable

To find the value of \( b \), take the square root of both sides of the equation. This will give two possible solutions because both positive and negative numbers squared give the same result.

Step 3 - Calculate the square roots

Calculate the square root of 144: \[ b = \pm \sqrt{144} \]. The square root of 144 is 12, so the solutions are \( b = 12 \) and \( b = -12 \).

Step 4 - State the solutions

So, the solutions to the quadratic equation \( b^2 = 144 \) are \( b = 12 \) and \( b = -12 \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\). They often pop up in various areas of algebra and geometry. In this exercise, you encountered a simplified quadratic equation: \(b^2 = 144\). Solving such equations involves finding the value of the variable—in this case, \b\—that makes the equation true. Let's break it down further.

Square Roots

Square roots are key to solving many algebraic equations, especially quadratics. A square root of a number \x\ is a value that, when multiplied by itself, yields \x\. For example, the square root of 144 is 12 because \(12 \times 12 = 144\).

Positive and Negative Solutions

In quadratic equations, every solution has a 'twin'—its positive and negative counterpart. This means that for any \x^2 = c\, you'll find both \sqrt{c}\ and \-\sqrt{c}\ as solutions. The initial example of \(b^2 = 144\) perfectly illustrates this concept.