Problem 19 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ a^{2}-2 a-3=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are $$a = -1, 3$$.

Step 1: Rewrite the equation

Rewrite the given equation like this: $$a^2 - 2a - 3 = 0$$

Step 2: Move the constant

Move the constant term to the right side of the equation: $$a^2 - 2a = 3$$

Step 3: Complete the square

Add the square of half of the coefficient of the linear term to both sides. In our case, the linear term is -2a, so half the coefficient is -1. Square it (-1)^2 = 1, then add it to both sides: $$(a^2 - 2a + 1) = 3 + 1$$ $$\Rightarrow (a-1)^2 = 4$$

Step 4: Take the square root

Take the square root of both sides. We must remember to take both positive and negative square roots: $$a-1 = \pm \sqrt{4}$$ $$a-1 = \pm 2$$

Step 5: Solve for the variable

Now, solve for the variable a, which will give us two solutions: $$a = 1 + 2 = 3$$ $$a = 1 - 2 = -1$$ The solutions are: $$a =-1, 3$$

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra that allows you to find the values of an unknown variable that make the equation true. These equations are typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The methods to solve these include factoring, using the quadratic formula, graphing, and completing the square, which is an algebraic technique particularly useful when the quadratic cannot be easily factored.

Completing the square involves transforming the equation into a perfect square trinomial on one side, thus making it easier to solve. In the exercise provided, we start with the equation \( a^2 - 2a - 3 = 0 \) and then manipulate it step by step, so the left side becomes a squared binomial. This method not only helps find the solutions for \( a \) but also deepens the understanding of the structure of quadratic equations.

Algebraic Methods

Algebraic methods are the diverse set of tools used to manipulate and solve algebraic equations. Completing the square is one of these methods and is particularly advantageous when the quadratic equation does not factor neatly or when you are dealing with coefficients that are not perfect squares.

To employ this technique, we adjust the equation to prepare it for creating a perfect square trinomial. One essential tip to remember is that the coefficient in front of the \( x^2 \) term needs to be 1 before you can complete the square. If it's not, you would first divide the entire equation by that coefficient. Another key aspect is to always balance the equation by doing the same operations on both sides. In our example, we moved the constant term to the right side and then added the square of half of the \( x \) coefficient to both sides to obtain a perfect square trinomial.

Square Root Method

The square root method is another way to solve quadratic equations, which is particularly effective after completing the square. Once you've transformed the quadratic into \((a - b)^2 = c\), where \( a \), \( b \), and \( c \) are real numbers, you can proceed to isolate \( a \) by taking the square root of both sides of the equation. Remember that when you take the square root of both sides, it introduces the possibility of two solutions, a positive and a negative root.

In the solved exercise, after obtaining \((a-1)^2 = 4\), we took the square root of both sides, which resulted in \(a-1 = \pm \sqrt{4}\). This gives us two possible solutions because \(\sqrt{4}\) can be both 2 and -2. Following this method allows us to easily find the final solutions for \( a \) by isolating and simplifying the equation.

Quadratic Formula Alternative

The quadratic formula provides an alternative solution to completing the square and is derived directly from the process of completing the square itself. It is given as \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\). While completing the square is a step-by-step approach that gives insight into the structure of the equation, the quadratic formula offers a more direct route to the solutions.

To use the formula, simply identify the coefficients from the original equation and substitute them into the formula. Despite its efficiency, many educators emphasize understanding completing the square, as it enhances algebraic reasoning and problem-solving skills. Moreover, the quadratic formula does not always provide the same level of intuitive understanding as manipulating the equation through completing the square.