Problem 9 - Elementary Algebra

Question

Solve each of the following quadratic equations using the method of extraction of roots. $$ (m-4)^{2}=15 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation (m-4)^2 = 15 by the method of extraction of roots and find the two possible values of m. Answer: The two possible values of m are: 1. m = 4 + √15 2. m = 4 - √15

Step 1: Write down the given equation

First, let's write down the given quadratic equation: $$(m-4)^2=15$$

Step 2: Take the square root of both sides

Next, take the square root of both sides to remove the perfect square term: $$\sqrt{(m-4)^2}=\sqrt{15}$$

Step 3: Simplify the square root

Now, simplify the left side as follows: $$m-4=\pm\sqrt{15}$$ Notice that there are two possible results, since the square root of a number squared gives both the positive and negative values of the original number.

Step 4: Solve for m

Finally, add 4 to both sides to solve for \(m\): $$m=4\pm\sqrt{15}$$ So the two solutions for \(m\) are: 1. \(m=4+\sqrt{15}\) 2. \(m=4-\sqrt{15}\)

Key Concepts

Extraction of Roots

The method of extraction of roots is a straightforward technique used to solve certain quadratic equations. It is particularly handy when dealing with equations that can be reduced to the form \(x^2 = a\), where \(x\) is a variable and \(a\) is a constant. The idea is to isolate the squared term on one side of the equation and then apply the square root to both sides. This process essentially reverses the squaring process and provides the value of \(x\).

Remember to consider both the positive and negative square roots, since squaring either positive or negative values will yield the same result. For example, in the given exercise \(m-4)^2=15\), after applying the square root, we obtain \(m-4=\pm\sqrt{15}\), which reflects the fact that both \(m = 4 + \sqrt{15}\) and \(m = 4 - \sqrt{15}\) are valid solutions.

This method is most effective when you're working with perfect squares or when you can easily convert an equation into a perfect square. It eliminates the need for factoring or completing the square, streamlining the solution process for quadratic equations that fit the criteria.

Square Root Method

The square root method is closely associated with the extraction of roots. It is a procedure used to solve quadratic equations that are in perfect square form or can easily be converted to one. The tactic here is to keep the equation balanced while freeing the variable from its squared state.

To apply this method, as shown in the exercise, the steps include:

• Isolating the perfect square term on one side of the equation.
• Taking the square root of both sides, which gives us two possibilities due to the nature of square roots.
• Simplifying the resulting expression to find the values of the variable that satisfy the equation.

When the equation does not include a perfect square initially, other techniques such as completing the square could be employed first to bring the equation to a form suitable for the square root method.

Perfect Square Trinomials

A trinomial is a type of polynomial that has three terms. Perfect square trinomials are a special set of polynomials that can be factored into a binomial squared, such as \(x + a)^2\) or \(x - a)^2\). They have the form \(x^2 + 2ax + a^2\) or \(x^2 - 2ax + a^2\), where the first and last terms are the squares of binomial terms, and the middle term is double the product of the two terms of the binomial.

If you expand the squares \(x + a)^2\) or \(x - a)^2\), you will get the respective perfect square trinomials, which confirms their patterns. Recognizing these patterns can be incredibly useful in solving quadratic equations, as they can be solved quite easily using the extraction of roots or the square root method once identified. In some cases, you may need to manipulate the equation slightly by adding or subtracting terms to both sides to create a perfect square trinomial.