Problem 93 - Algebra 1

Question

Solve the equation. $$x^{2}=\frac{9}{25}$$

Step-by-Step Solution

Answer

The solutions of the given equation are: \(x= \frac{3}{5}\) or \(x= -\frac{3}{5}\)

Step 1: Analyze the equation

The given equation is \(x^{2} = \frac{9}{25}\). It is a typical quadratic function where the variable \(x\) is squared.

Step 2: Take the square root of both sides

To find the value of \(x\), let's take the square root of both sides of the equation. This will give us two solutions, one positive, and one negative, since both a negative and positive number squared would yield the original number. So, we have \(x= \sqrt{\frac{9}{25}}\) or \(x= -\sqrt{\frac{9}{25}}\)

Step 3: Calculate the square root of the fraction

Calculating the square root of the given fraction, we have \(x= \sqrt{\frac{9}{25}} = \frac{3}{5}\) or \(x= -\sqrt{\frac{9}{25}} = -\frac{3}{5}\)

Key Concepts

Square Root

The square root is a mathematical operation that is essentially the opposite of squaring a number. When you square a number, you multiply it by itself. The square root, on the other hand, identifies which number, when squared, gives you the original value. For instance, since 3 x 3 equals 9, the square root of 9 is 3.
In mathematical notation, the square root of a number can be represented by the radical symbol, √. For example, the square root of 9 is written as \( \sqrt{9} \). However, it's essential to remember that both positive and negative values can be square roots, because both 3 x 3 and -3 x -3 equal 9.

• Positive square root (principal): \( \sqrt{9} = 3 \)
• Negative square root: \( -\sqrt{9} = -3 \)

In problems like the equation \( x^2 = \frac{9}{25} \), taking the square root gives two potential answers: a positive and a negative result, making the solutions \( x = \frac{3}{5} \) and \( x = -\frac{3}{5} \).

Solving Quadratic Equations

Quadratic equations often come in the standard form \( ax^2 + bx + c = 0 \), but sometimes they appear as simple cases like \( x^2 = n \). The equation from our example, \( x^2 = \frac{9}{25} \), is a quadratic equation where the term with \( b \) and constant \( c \) are zero.
When solving these equations, taking the square root of both sides is a common method; it's straightforward when other terms don't complicate the equation. Therefore, you find \( x \) by calculating: \( x = \pm \sqrt{n} \).

• First, isolate \( x^2 \) if necessary so that the equation is \( x^2 = n \).
• Then, calculate \( x = \sqrt{n} \) and \( x = -\sqrt{n} \).

Whether the solution involves more complex techniques like factoring, completing the square, or using the quadratic formula, these methods rely on similar principles: finding the value of \( x \) that satisfies the equation under different scenarios.

Rational Numbers

Rational numbers are numbers that can be expressed as the fraction of two integers, \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). For instance, \( \frac{9}{25} \) and \( 3 \) are rational numbers.
The square root of a rational number often results in another rational number. This is evident from our example where \( \sqrt{\frac{9}{25}} \) results in \( \frac{3}{5} \), which is a rational number because both 3 and 5 are integers.

• To find the square root of a rational number \( \frac{a}{b} \), find separately: \( \sqrt{a} \) and \( \sqrt{b} \), provided they are perfect squares.
• If \( a = m^2 \) and \( b = n^2 \), then \( \sqrt{\frac{a}{b}} = \frac{m}{n} \)

Understanding rational numbers is key when working with fractions, as they appear extensively in solving equations and working with algebra.