Problem 55 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ 2 a^{2}+8 a=-12 $$

Step-by-Step Solution

Answer

The solutions are \(a = -2 \pm i \sqrt{2}\).

Step 1: Move Constant to Right Side

Start by moving the constant term from the left side of the equation to the right side. The equation is: \(2a^2 + 8a = -12\). Divide the equation by 2 to simplify: \(a^2 + 4a = -6\).

Step 2: Calculate the Completing Square Term

Take half of the coefficient of the linear term (which is 4) and square it to get the completing square term. Half of 4 is 2, and \(2^2 = 4\).

Step 3: Add Completing Square Term

Add the calculated square term to both sides of the equation to complete the square: \(a^2 + 4a + 4 = -6 + 4\). This simplifies to \((a + 2)^2 = -2\).

Step 4: Solve the Square Root Equation

Take the square root of both sides of the equation to solve for \(a\): \(a + 2 = \pm \sqrt{-2}\).

Step 5: Simplify to Get Final Answers

Subtract 2 from both sides to solve for \(a\): \(a = -2 \pm i \sqrt{2}\). Thus, the solutions are \(a = -2 + i \sqrt{2}\) and \(a = -2 - i \sqrt{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the unknown variable. These types of equations are fundamental in algebra and appear frequently in various fields of mathematics and science.
They have a characteristic "U" shape graph known as a parabola. The solutions or 'roots' of a quadratic equation determine where the parabola intersects the x-axis.We can solve quadratic equations using several methods, each with its unique approach:

• Factoring: Breaking down the expression into simpler binomial terms.
• Quadratic Formula: Employing the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find solutions.
• Completing the Square: Rewriting the equation in a form that allows extraction of roots directly.

The method of completing the square is particularly useful when the quadratic does not factor neatly. It sets up the equation to express it as a binomial square, making roots apparent.

Complex Numbers

Complex numbers introduce an extension to our conventional number system. They are numbers of the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, defined by \(i^2 = -1\). This expands our ability to solve equations that involve negative square roots.
In our equation, we ended up with \(a = -2 \pm i \sqrt{2}\). This result includes complex numbers due to the square root of a negative number, specifically \(-2\).
To understand complex solutions better, remember:

• The imaginary unit \(i\) exists to provide solutions to equations without real roots.
• Complex numbers often appear as conjugate pairs, denoted by \(a + bi\) and \(a - bi\).
• They are plotted on a plane with horizontal real and vertical imaginary axes, known as the complex plane.

Mastering complex numbers is crucial, as they frequently emerge in advanced mathematics and engineering applications.

Solving Equations

Solving equations involves deriving values for the unknown variable that make the equation hold true. This process can be simple or complex, depending on the equation and methods used.
For our quadratic equation, solving involves transforming the equation into a form where the variable \(a\) is isolated. We applied the "completing the square" method to make the equation manageable.
Steps to solve by completing the square include:

• Rearrange the Equation: Move the constant to the right side to isolate the terms involving the variable.
• Simplify if Needed: Divide if needed to make the leading coefficient 1.
• Add Completing Square Term: Square half the linear coefficient and add this square to both sides.
• Square Root Both Sides: Simple algebraic manipulation and using properties of square roots.
• Solve for the Variable: Isolate the variable to get the solutions.

Each step transforms the equation closer to a form that allows straightforward determination of the variable.

Algebraic Manipulation

Algebraic manipulation is the process of transforming equations using well-defined algebraic operations, which make it easier to solve or simplify expressions.
Completing the square is one such manipulation technique applied to quadratic expressions like \(2a^2 + 8a = -12\). Here, we rewrite the expression into a perfect square plus a constant, which simplifies the solution process.Steps include:

• Simplification: Divide the equation to make the leading term a simpler form (e.g., coefficient of \(a^2\) becomes 1).
• Rebalancing the Equation: Ensure whatever changes you make to one side are mirrored on the other to maintain equality.
• Utilizing Mathematical Properties: Apply properties such as factoring, distributive property, and square root property.

Mastering algebraic manipulation is essential in resolving complex expressions and making challenging problems manageable.