Problem 15 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ a^{2}=9 $$

Step-by-Step Solution

Answer

Answer: The values of 'a' in the given quadratic equation are 3 and -3.

Step 1: Identify the given equation

The given quadratic equation is: $$ a^{2}=9 $$

Step 2: Extract the square root of both sides

To find the value of 'a', we need to take the square root of both sides of the equation. It results in: $$ \sqrt{a^{2}}=\sqrt{9} $$

Step 3: Simplify the equation

Simplifying the equation results in: $$ a=\pm3 $$

Step 4: Write the final solution

The solutions for 'a' in the given quadratic equation are: $$ a = 3, a = -3 $$

Key Concepts

Quadratic Equations

Quadratic equations are at the heart of algebra, forming a fundamental part of high school mathematics. They are second-order polynomials, typically presented in the form \(ax^2 + bx + c = 0\), where \('a'\), \('b'\), and \('c'\) are constants, and \('a'\) is not equal to zero. Solving these equations involves finding the value(s) of \('x'\) that make the equation true. The extraction of roots is a method used for special quadratic equations where the equation is a perfect square and can be written as \( x^2 = k \), where \(k\) is a non-negative real number. This method simplifies solving as it directly uses the property that a square root of a squared number can yield both positive and negative solutions.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for square root is \('\sqrt{}'\). For example, the square root of 9 is 3, because \(3 * 3 = 9\). In algebra, when you take the square root of both sides of an equation such as \(x^2 = k\), you will get \(x = \pm\sqrt{k}\). The '±' symbol indicates that there are usually two solutions to a quadratic equation: one positive and one negative. Understanding square roots is essential when dealing with quadratic equations because it is one of the methods to solve them when they are in the perfect square form.

Algebraic Solutions

Algebraic solutions involve finding the values that satisfy an equation. In the context of quadratic equations, this could mean using various methods such as factoring, completing the square, using the quadratic formula, or extraction of roots, depending on the form of the equation at hand. Algebraic solutions must satisfy the original equation, and when dealing with square roots, we often get two solutions. This is because squaring a positive or negative number results in the same square value. Therefore, algebraic solutions involve a logical series of steps that guide you from the initial problem to the final answers, considering all possible solutions.

Elementary Algebra

Elementary algebra is the foundational branch of algebra that deals with the basics of algebraic expressions, equations, and functions. In this domain, one learns to manipulate algebraic expressions and solve equations involving variables. For instance, in solving the equation \(a^2 = 9\) through extraction of roots, one employs elementary algebraic techniques such as taking square roots and simplifying expressions. Thus, extraction of roots as a method to solve quadratic equations is an application of elementary algebra. Being proficient in elementary algebra is vital for anyone hoping to understand more complex mathematical concepts and various applications in science and engineering.