Problem 20 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ x^{2}=81 $$

Step-by-Step Solution

Answer

Answer: The values of x that satisfy the given quadratic equation are x = 9 and x = -9.

Step 1: Identify the given equation

Here, the given quadratic equation is $$x^2 = 81$$.

Step 2: Apply the method of extraction of roots

Take the square root of both sides of the equation. Remember that taking the square root of a number has two possible answers; one positive (+) and one negative (-). $$\sqrt{x^2} = \pm\sqrt{81}$$ $$x= \pm\sqrt{81}$$

Step 3: Calculate the square root

Now, we calculate the square root of 81, remembering to consider both positive and negative possibilities for the answer. $$x = \pm9$$

Step 4: Write the final answer

The solution of the equation $$x^2 = 81$$ using method of extraction of roots is: $$x = 9, -9$$

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental component of algebra and come in the standard form of \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations are called quadratic because 'quad' means square, and the variable gets squared (i.e., \( x^2 \)).

It's important for students to understand that any equation of this form can have up to two real solutions, and these solutions are where the parabola, the graph of the quadratic equation, intersects the x-axis.

Why Two Solutions?

This is because the squared term \( x^2 \), inherently leads to two values of 'x' that satisfy the equation - one positive and one negative. Thus, solving quadratic equations is about finding these 'x' values that make the equation true.

Square Roots

The square root of a number is another number, which when multiplied by itself, gives the original number. For example, since \( 9 \times 9 = 81 \), we say the square root of 81 is 9. It's denoted as \( \sqrt{81} = 9 \).

A key point for students is that every positive number actually has two square roots: a positive and a negative root, because both \( 9 \times 9 \) and \( -9 \times -9 \) equal 81. This is why we write \( \sqrt{81} = \pm9 \). The ± symbol indicates that there are two roots: \( +9 \) and \( -9 \). Always remember, when dealing with square roots, to consider both the positive and negative possibilities.

Solving Quadratic Equations

Solving quadratic equations can be approached using several methods, including factorization, completing the square, using the quadratic formula, or the method of extraction of roots, as seen in the given exercise.

The extraction of roots method involves isolating the \( x^2 \)-term on one side of the equation and then taking the square root of both sides. This method works well when the equation can be easily rewritten in the form of \( x^2 = \text{some number} \).

Applying Extraction of Roots

In our example, \( x^2 = 81 \), we take the square root of both sides and remember that there are always two potential solutions, \( x = \sqrt{81} \) or \( x = -\sqrt{81} \), leading to \( x = 9 \) and \( x = -9 \). This method is straightforward and efficient when applicable, making it a valuable tool for solving certain types of quadratic equations.