Problem 31 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ x^{2}-10=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation using the method of extraction of roots: $$x^2-10=0$$ Answer: The solutions for the given quadratic equation are $$x=\sqrt{10}$$ or $$x= -\sqrt{10}$$.

Step 1: Isolate the variable term

To isolate the variable term (x^2), we will add 10 to both sides of the equation: $$x^2-10+10=0+10$$ which simplifies to $$x^2=10$$.

Step 2: Take the square root of both sides

To find the value(s) of x, we will take the square root of both sides of the equation. Remember that when taking the square root of both sides, we should consider both the positive and negative square roots:$$\sqrt{x^2}=\pm\sqrt{10}$$ Simplifying, we get:$$x=\pm\sqrt{10}$$ So, the solutions for the given quadratic equation are:$$x=\sqrt{10}$$ or $$x= -\sqrt{10}$$.

Key Concepts

Extraction of Roots

The extraction of roots is a simple yet powerful method to solve specific quadratic equations. It is especially useful when your equation is in the form of a perfect square.
To apply this method, we need to isolate the squared term on one side of the equation. For example, consider the quadratic equation \(x^2 - 10 = 0\). We start by moving the constant to the other side to have \(x^2 = 10\). This isolation makes it possible to use the square root method effectively.

• Ensure the squared term \(x^2\) is by itself on one side of the equation.
• Move any constants to the opposite side of the equation to simplify solving.

After simplifying to \(x^2 = 10\), we proceed to use the square root method. This takes us to our next section about learning how to take square roots properly.

Square Root Method

The square root method is an essential tool for solving quadratic equations, where you take the square root of both sides. This method is applied after isolating the square term of the equation, such as identifying \(x^2 = 10\).
You must remember to consider both the positive and negative roots when using this method because squaring a number can result in the same value whether the original number was positive or negative. For instance:

• \(x = \pm \sqrt{10}\) represents two potential values for \(x\): \(\sqrt{10}\) and \(-\sqrt{10}\).
• This ensures all possible solutions for \(x\) are included.

The extraction of roots through the square root method is straightforward, but understanding that it gives two symmetric solutions across the x-axis is crucial for grasping the full concept.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. There are several methods to solve a quadratic equation: by factoring, using the quadratic formula, completing the square, or, as we explored, through extraction of roots. Each method can be useful depending on the structure of the equation you are working with.

• For equations like \(x^2 = 10\), the square root method is the simplest.
• Always check both solutions obtained because quadratic equations often have two roots.

Remember, solving quadratics is about opening up multiple paths to reach the correct conclusion. Here, the method of extracting roots allows you to directly solve equations that can be simplified to contain perfect square terms. Understanding when to use each method is key to mastering quadratic equations.