Problem 77 - Elementary Algebra

Question

77\. Problem: Solve: \(m^{2}=125\) Incorrect Answer: \(m^{2}=125\) $$ \text { Incorrect Answer: } \begin{aligned} m^{2} &=125 \\ m &=\sqrt{125} \text { or } m=-\sqrt{125} \\ m &=\pm \sqrt{125} \end{aligned} $$

Step-by-Step Solution

Answer

m = \pm 5\sqrt{5}

Step 1: Identify the problem

The equation given is: \(m^{2} = 125\). The goal is to find the value(s) of \(m\) that satisfy this equation.

Step 2: Apply the square root

To solve for \(m\), take the square root of both sides of the equation. Remember that taking the square root of both sides involves considering both the positive and negative roots. Hence, \(m = \pm \sqrt{125}\).

Step 3: Simplify the square root

Simplify the expression \(\sqrt{125}\). Note that \(125\) can be factored into \(25 \times 5\), so \(\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}\).

Key Concepts

Square Root

Understanding the concept of the square root is fundamental in solving quadratic equations. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3^2 = 9\). When solving equations like \(m^2 = 125\), taking the square root of both sides helps us find the values of \(m\). It's important to note that a square root operation can yield both a positive and a negative value. This is because \[ (\text{positive} \times \text{positive}) = (\text{negative} \times \text{negative})\]. Hence, for \[m^2 = 125\], both \[ \sqrt{125} \text{and} -\sqrt{125}\] are valid solutions.

Positive and Negative Roots

When solving quadratic equations such as \[ m^2 = 125\], it’s crucial to account for both positive and negative roots. This means we consider both \[ \sqrt{125}\] and \[ -\sqrt{125}\] as solutions. This dual consideration stems from the property of squaring, where both a positive and a negative number squared result in the same positive product. Neglecting either root results in an incomplete solution. Always express the solution to such equations in the form \[ m = \pm \sqrt{125} \] indicating both possibilities.

Simplifying Square Roots

Simplifying square roots involves expressing the root in its simplest form, making it easier to understand and use. For example, to simplify \[ \sqrt{125}\], we start by factoring the number. Recognize that \[ 125 = 25 \times 5\]. Since \[ \sqrt{25} = 5\], we can rewrite \[ \sqrt{125} = \sqrt{25 \times 5} = 5 \sqrt{5} \]. This simplified expression, \[ 5 \sqrt{5}\], is more manageable and preferred in mathematical solutions. Simplifying helps in recognizing the relationships between numbers and often aids further calculations.

Factoring

Factoring is a critical process in simplifying square roots and solving quadratic equations. By breaking down a number into its prime factors, we can simplify the expressions more effectively. For instance, to solve \[ m^2 = 125\] and simplify \[ \sqrt{125}\], we factor 125 as \[ 25 \times 5\] because 25 is a perfect square. Identifying perfect squares within the factors helps in simplifying the square root. For complex quadratic equations, factoring the quadratic expression directly can provide solutions quickly. Begin by identifying pairs of factors and then simplify the square roots judiciously.