Problem 15 - Elementary Algebra

Question

For the following problems, solve the equations. $$ a^{2}+6 a+8=0 $$

Step-by-Step Solution

Answer

Question: Find the two possible values of 'a' that satisfy the equation $$a^{2}+6a+8=0$$. Answer: The two possible values of 'a' that satisfy the given equation are $$a = -2$$ and $$a = -4$$.

Step 1: Identify the coefficients

The given equation is $$a^{2}+6a+8=0$$. We can see that the coefficients are $$a = 1$$, $$b = 6$$, and $$c = 8$$.

Step 2: Plug the coefficients into the quadratic formula

Now, we will plug the coefficients into the quadratic formula: $$ a = \frac{-6 \pm \sqrt{6^2 - 4\cdot1\cdot8}}{2\cdot1} $$

Step 3: Simplify the equation

Let's simplify the equation: $$ a = \frac{-6 \pm \sqrt{36 - 32}}{2} $$ Now calculate the discriminant ($$b^2 - 4ac = 36 - 32$$): $$ a = \frac{-6 \pm \sqrt{4}}{2} $$

Step 4: Solve for two possible values of 'a'

As we have a square root, there will be two possible values for $$a$$ depending on whether we use the positive or the negative square root. Let's solve for both: For the positive square root: $$ a_1 = \frac{-6 + 2}{2} = \frac{-4}{2} = -2 $$ For the negative square root: $$ a_2 = \frac{-6 - 2}{2} = \frac{-8}{2} = -4 $$ The two possible values of $$a$$ that satisfy the given equation are $$a = -2$$ and $$a = -4$$.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful method used for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula simplifies the process of finding the roots, or solutions, of the quadratic equation. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a breakdown of the formula:

• \(a\), \(b\), and \(c\) are the coefficients of the terms \(x^2\), \(x\), and the constant term, respectively.
• The \(\pm\) symbol means you'll calculate two separate solutions: one with \(+\) and one with \(-\).
• The term \(\sqrt{b^2 - 4ac}\) is crucial as it determines the nature of the roots, which we'll explore next.

The quadratic formula provides an efficient means to find the solutions directly without the need for factoring or graphing. It is therefore especially useful when the equation is complex or unfactorable by simple inspection.

Discriminant

The discriminant in a quadratic equation is found within the quadratic formula. It is the part under the square root symbol: \(b^2 - 4ac\). The discriminant indicates how many and what type of solutions exist:

• If \(b^2 - 4ac > 0\): The equation has two distinct real roots since the square root of a positive number is a real number.
• If \(b^2 - 4ac = 0\): The equation has exactly one real root, also known as a double root, because the square root of zero is zero.
• If \(b^2 - 4ac < 0\): The equation has no real roots; instead, there are two complex roots, which occur as complex conjugates.

In our problem, the discriminant is calculated as \(36 - 32 = 4\), which is positive. Thus, we have two distinct real solutions, aligning with our solved solutions of \(a = -2\) and \(a = -4\). Understanding the discriminant helps in predicting the nature of the solutions even before solving them fully.

Solving Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. In our example, the equation is \(a^2 + 6a + 8 = 0\). Here, we used the quadratic formula to solve for \(a\) because the equation did not easily factorize at a glance.
Starting with identifying the coefficients as \(a = 1\), \(b = 6\), and \(c = 8\), we plugged these into the quadratic formula:\[a = \frac{-6 \pm \sqrt{36 - 32}}{2(1)}\]Simplifying, we found the roots by evaluating the square roots and performing basic arithmetic:

• Using the positive square root gives \(a_1 = -2\).
• Using the negative square root gives \(a_2 = -4\).

Successful solving involves not only calculation but also understanding the method being applied. This blends analytical thinking with arithmetic skills, ensuring you achieve correct solutions in a structured way. Working through each step carefully ensures the solutions derived are accurate and verifiable.