Problem 15 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ a^{2}+4 a+7=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(a = -2 + \sqrt{3}i\) and \(a = -2 - \sqrt{3}i\).

Step 1: Setup the given equation

The given equation is: $$ a^{2}+4a+7 = 0 $$

Step 2: Move the constant term to the right side

Subtract 7 from both sides of the equation to isolate the constant term on the right side: $$ a^2 + 4a = -7 $$

Step 3: Add a term to both sides to complete the square

We need to add a term to both sides to allow us to rewrite the left side as a perfect square. To find that term, take half of the coefficient of the linear term (4), square it, and add it to both sides: $$ a^2 + 4a + (4/2)^2 = -7 + (4/2)^2 $$

Step 4: Simplify the equation and rewrite in squared form

Simplify the equation and rewrite the left side as a perfect square: $$ a^2 + 4a + 4 = -7 + 4 $$ $$ (a+2)^2 = -3 $$

Step 5: Solve for the variable a

Take the square root of both sides, and solve for a. Remember to include both positive and negative square roots: $$ \sqrt{(a+2)^2} = \pm\sqrt{-3} $$ $$ a+2 = \pm\sqrt{-3} $$ Now, subtract 2 from both sides to isolate a: $$ a = \pm\sqrt{-3} - 2 $$

Step 6: Write the solution as a complex number

Since the square root of a negative number is an imaginary number, express the solution in the form of a complex number: $$ a = -2 \pm \sqrt{3}i $$ So, the solutions to the given equation are \(a = -2 + \sqrt{3}i\) and \(a = -2 - \sqrt{3}i\).

Key Concepts

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, generally expressed in the form \( ax^2 + bx + c = 0 \). It comprises three terms:

• \( ax^2 \) is the quadratic term, which includes \( x \) raised to the power of 2.
• \( bx \) is the linear term, involving \( x \) to the first power.
• \( c \) is the constant term, a number without an associated variable.

The task of solving a quadratic equation is to find the values of \( x \) (or another variable like "a" in different problems) that make the equation true.
In the context of completing the square to solve a quadratic equation, we aim to transform the equation into a perfect square trinomial, which can be factored easily. This method involves isolating the variable terms and adjusting the equation so we neatly "complete" the square.
"Completing the square" is a very flexible tool not only for solving quadratics, but also in calculus where it helps integrate complex polynomial expressions.

Complex Numbers in Solving Quadratics

Complex numbers extend the concept of ordinary numbers in the real number system and allow solutions to equations that don't have real solutions. They are expressed in the form \( a + bi \), where:

• \( a \) is the real part.
• \( bi \) is the imaginary part, with \( i \) being the imaginary unit, defined as \( \sqrt{-1} \).

When we solve quadratic equations using the method of completing the square, particularly if the equation results in a term like the square root of a negative number, we face complex numbers.
This happens because there is no real number the square of which is negative. Hence, in such cases, the solutions to the quadratic equation come in conjugate pairs. For example, from the equation solution \( a = -2 \pm \sqrt{3}i \), we see two complex solutions: one with \( + \) and the other with \( - \) before the imaginary term.

Step-by-Step Process of Solving Equations

Solving equations, especially quadratics, involves a systematic process that guides us to find the unknown variable's value(s). Here's how it typically unfolds:

• **Setup the Equation**: Ensure the equation is in its standard form and identify the quadratic, linear, and constant terms.
• **Rearrange Terms**: Move the constant to one side, so you can complete the square on the other.
• **Complete the Square**: Transform the equation's left side into a perfect square trinomial by adding a calculated term to both sides. This generally involves halving the linear coefficient, squaring it, and then applying this to modify the equation.
• **Solve for the Variable**: Once you have a perfect square, take the square root of both sides to simplify the equation.
• **Isolate the Variable**: Rearrange the equation to solve for the unknown, taking into account that you will now have real and/or imaginary components if the square root involves a negative.

The beauty of this method lies in its logical clarity, allowing us to discover not only real solutions but complex ones as well, enriching our understanding of mathematics beyond real number operations.