Problem 51 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ 3\left(x^{2}+1\right)=2(x+7) $$

Step-by-Step Solution

Answer

Based on the step-by-step solution provided, create a short answer question: Question: Given the quadratic equation \((a+10)^2 = 1\), solve for the variable \(a\) using the method of extraction of roots. Answer: After extracting the square root from both sides and isolating the variable \(a\), we find two possible solutions: \(a = -9\) and \(a = -11\).

Step 1: Rewrite the equation

We will begin by rewriting the equation in a more suitable form: $$ (a+10)^2 = 1 $$

Step 2: Extract square root from both sides

In this step, we will extract the square root from both sides of the equation. Remember that when extracting the square root, we consider both positive and negative roots: $$ \sqrt{(a+10)^2} = \pm\sqrt{1} $$

Step 3: Simplify the equation

Now we will simplify the equation by removing the square from the left side and finding the square root of 1 on the right side: $$ a + 10 = \pm 1 $$

Step 4: Isolate variable \(a\)

Since we have two different cases (positive and negative square root), we will find the corresponding values of \(a\) for each case: Case 1: Positive square root $$ a + 10 = 1 \Rightarrow a = 1 - 10 \Rightarrow a = -9 $$ Case 2: Negative square root $$ a + 10 = -1 \Rightarrow a = -1 - 10 \Rightarrow a = -11 $$

Step 5: Present the final solution

Now we have two different values of \(a\), so the final solution will be: $$ a = -9, -11 $$

Key Concepts

Extraction of Roots

The extraction of roots is a fundamental concept in solving quadratic equations. When you see an equation involving a squared term, such as \[(a+10)^2 = 1,\]the first step is to extract the square root. Extracting the roots means you will take the square root of both sides of the equation. Doing this will help in solving the equation easily.

However, remember that every positive number has two square roots: a positive and a negative. For example, the square root of 1 is either 1 or -1. Thus, when you extract roots from both sides, write it as\[\sqrt{(a+10)^2} = \pm \sqrt{1}. \]
This means you have to solve for two possibilities, leading to potentially two different solutions. Extraction of roots is an effective method because it helps to simplify complex squared equations into manageable linear forms.

Solving Equations

In solving equations, especially ones involving squares, extracting roots plays a crucial role. After extracting the square roots, the equation simplifies from \[(a+10)^2 = 1 \]to \[a+10 = \pm 1.\]Now, these two simple equations are easier to analyze and solve.

Here's how solving each equation works:

• For the positive root, \(a + 10 = 1\)
• For the negative root, \(a + 10 = -1\)

You then isolate the variable \(a\) to solve for its value:

• In the first case: \(a + 10 = 1\) becomes \(a = 1 - 10\).
• In the second case: \(a + 10 = -1\) becomes \(a = -1 - 10\).

In both cases, simple arithmetic leads you to the solutions \(a = -9\) and \(a = -11\). Solving equations requires evaluating each possible solution carefully to ensure they are correct within the context of the problem.

Square Roots

Understanding square roots is vital when dealing with quadratic equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 1 is 1 because \(1 \times 1 = 1\). However, the square root can also be -1, since \((-1) \times (-1) = 1\) as well. This dual possibility is why we often write the square root of a number as \(\pm\) something.

### Importance of Considering Both RootsWhen solving equations that involve squares, always remember to consider both the positive and negative roots. It guarantees that you account for every potential solution. This is especially important in exact sciences where missing a possibility can lead to incorrect conclusions.

Getting comfortable with the concept of square roots:

• Visualize square roots as the side length of a square whose area is represented by the number.
• Practice finding square roots of simple numbers to build your intuition.

As you handle more complex equations, this understanding will aid in solving them quickly and accurately.