Problem 42 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ x^{2}+6 x+8=-x-2 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation using the extraction of roots method: $$4x^2 - 64 = 0$$ Answer: The two solutions for the given equation are \(x = 4\) and \(x = -4\).

Step 1: Divide the equation by the coefficient of the squared term

To make it easier to work with, let's first divide the entire equation by the coefficient of the squared term (4): $$ \frac{4x^2}{4} - \frac{64}{4} = \frac{0}{4} $$ This simplifies to: $$ x^2 - 16 = 0 $$

Step 2: Isolate the squared term

Now, we need to isolate the squared term by moving the constant to the other side of the equation: $$ x^2 = 16 $$

Step 3: Use the square root property

The square root property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). So, we can use the square root property to find the two solutions for x: $$ x = \pm \sqrt{16} $$ Now we can calculate the square root of 16: $$ x = \pm 4 $$ So the two solutions to the equation are \(x = 4\) and \(x = -4\).

Key Concepts

Extraction of Roots

In mathematics, the process of solving quadratic equations often involves 'extraction of roots.' This method essentially aims to "solve" the algebraic equation by finding the values of the variable that make the equation true. When you have an equation like \(4x^2 - 64 = 0\), extracting roots is an efficient way to identify the solutions.

To begin, it's important to simplify the equation to make the squared term easy to manage. This involves dividing the entire equation by the number that is multiplying the squared variable, typically referred to as the coefficient. Here, you would divide everything by 4:

• \(\frac{4x^2}{4} - \frac{64}{4} = \frac{0}{4}\)
• Which simplifies to \(x^2 - 16 = 0\)

After simplifying, you can proceed to isolate the squared term, which is part of the next section on isolation of variable. Remember, by simplifying the equation, you are preparing to extract the roots, which means finding the original numbers that were squared to result in the given terms.

Square Root Property

The square root property is a crucial principle used in the process of solving quadratic equations. It states that for any real number \(a\), if \(x^2 = a\), then \(x = \pm \sqrt{a}\). This property allows us to systematically find solutions to quadratic equations without needing to factor them.

When you have isolated the squared variable (for instance, \(x^2 = 16\)), applying the square root property helps you quickly find the potential values of \(x\). In practice, this process looks like:

• Recognizing that \(x^2 = 16\), which means \(x = \pm \sqrt{16}\)
• Calculating the square root to determine that \(x = \pm 4\)

This results in two possible solutions, namely \(x = 4\) and \(x = -4\). It's vital to include both the positive and negative solutions because squaring either result will give the original term (since \((4)^2 = 16\) and \((-4)^2 = 16\)).

Using the square root property is a powerful tool because it provides simplicity and speed when working through these kinds of quadratic equations.

Isolation of Variable

In solving equations, particularly quadratics, isolation of variable is a fundamental step that simplifies the problem. Isolating the variable means getting it on one side of the equation by itself, often so you can apply further methods or properties, such as the square root property we discussed earlier.

Look at the simplified equation \(x^2 - 16 = 0\). To isolate \(x^2\), you need to move any constants to the other side of the equation:

• Add 16 to both sides: \(x^2 = 16\)

This isolated squared term is now ready for the next phase of problem-solving. The beauty of isolating the variable is that it helps in reducing complex equations to a more manageable form. This makes it easier to apply operations like taking square roots.

So, isolation of the variable prepares the equation for the next logical step in finding the solution. Whether you're extracting roots or using the quadratic formula in more complicated equations, isolating the variable is essential for clarity and ease of solving.