Problem 31 - Beginning and Intermediate Algebra

Question

Solve each equation. $$36=25 n^{2}$$

Step-by-Step Solution

Answer

The solution to the equation \(36 = 25n^2\) is \(n = \frac{6}{5}\) and \(n = -\frac{6}{5}\).

Step 1: Isolate the n-squared term

Divide both sides of the equation by 25 in order to isolate the n-squared term: \[ \frac{36}{25} = n^2 \]

Step 2: Take the square root of both sides

To find the value of n, we need to take the square root of both sides of the equation: \[ \sqrt{\frac{36}{25}} = \sqrt{n^2} \]

Step 3: Simplify and solve for n

Now, we simplify the equation and find the value of n: \[ \frac{\sqrt{36}}{\sqrt{25}} = n \] Since 36 has two square roots, ±6, and 25 has one square root, 5, we have: \[ n = \frac{±6}{5} \] So, there are two possible values for n: \[ n = \frac{6}{5}, n = -\frac{6}{5} \]

Key Concepts

Solving Equations

To solve an equation like the one given, we need to find the values of the variable that make the equation true. The equation provided is a basic example of a quadratic equation: \(36 = 25n^2\). Our goal is to isolate the variable \(n\). This means we want \(n\) on one side of the equation and everything else on the other.

• Start by adjusting the equation so \(n^2\) is by itself. In this case, divide both sides by 25.
• Mathematically, this step rearranges our equation to \(\frac{36}{25} = n^2\).

When solving equations, each step you take is intended to simplify down to the variable you are interested in. Keep your operations equal on both sides to retain a balanced equation, leading you steadily to the solution. Adjusting equations involves basic operations like addition, subtraction, multiplication, division, and other algebraic manipulations.

Square Roots

The next step involves understanding square roots. A square root asks the question, "What number multiplied by itself gives me this number?" For the equation \(n^2 = \frac{36}{25}\), solving for \(n\) requires us to find the square root of both sides.

• The square root of \(n^2\) is \(n\), because \(n \times n = n^2\).
• For the fraction \(\frac{36}{25}\), take the square root of each part separately: \(\sqrt{36} = 6\) and \(\sqrt{25} = 5\).

This step simplifies the equation to \(n = \pm \frac{6}{5}\). Using square roots introduces the concept of two solutions: a positive and a negative outcome. In quadratic equations, this is because both \(x\) and \(-x\) would give the original squared value when squared.

Algebraic Expressions

The given problem also highlights working with algebraic expressions, where understanding how to manipulate and simplify expressions is crucial. An algebraic expression includes variables, constants, and operations which you can rearrange and solve.In our example \(\frac{36}{25} = n^2\), the expression on the left side is a fraction. Working with fractions includes:

• Understanding numerators and denominators, like 36 (numerator) and 25 (denominator).
• Using square roots effectively by recognizing that each part can be separately simplified: \(\frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}\).

When solving algebraic expressions, practice breaking down each term and operation. Each small step clarifies the path to the final variable solution. Keep in mind that algebra is like a puzzle, and each piece fits together logically to produce the answer.