Problem 53 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ y^{2}+2 y+2=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = -1 \pm i \).

Step 1: Move the constant term

First, let's rewrite the equation to isolate the constant on one side. We have: \[ y^2 + 2y + 2 = 0 \] Subtract 2 from both sides to get: \[ y^2 + 2y = -2 \]

Step 2: Complete the square

To complete the square, we need to add a certain number \( \left( \left(\frac{b}{2}\right)^2 \right) \) to both sides of the equation, where \( b \) is the coefficient of \( y \). Here, \( b = 2 \), so: \[ \left( \frac{2}{2} \right)^2 = 1 \] Therefore, add 1 to both sides:\[ y^2 + 2y + 1 = -2 + 1 \]This becomes: \[ (y + 1)^2 = -1 \]

Step 3: Take the square root of both sides

The next step is to take the square root of both sides of the equation: \[ \sqrt{(y + 1)^2} = \pm \sqrt{-1} \] Thus, we have: \[ y + 1 = \pm i \]

Step 4: Solve for y

Finally, solve for \( y \) by subtracting 1 from both sides: \[ y = -1 \pm i \]So, the solutions are: \[ y = -1 + i \] and \[ y = -1 - i \]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This technique transforms the quadratic equation into a perfect square trinomial, making it easier to solve.
To complete the square, follow these steps:

• Ensure the coefficient of \( y^2 \) is 1. If it isn't, divide the entire equation by the coefficient of \( y^2 \).
• Move the constant \( c \) to the other side of the equation.
• Add \( \left( \frac{b}{2} \right)^2 \) to both sides to form a perfect square trinomial on one side.
• Rewrite the trinomial as the square of a binomial.

In our example, we started with the equation \( y^2 + 2y + 2 = 0 \). By moving the constant 2 and completing the square, we formed \((y + 1)^2 = -1\). This prepares the equation for taking the square root in the next step.

Complex Numbers

When solving certain quadratic equations, you might encounter the square root of a negative number. This is where complex numbers come into play. Complex numbers extend the idea of traditional numbers and include an imaginary unit, denoted as \( i \), where \( i^2 = -1 \).

Complex numbers have a real part and an imaginary part, written in the form \( a + bi \). Here, \( a \) is the real part, and \( b \) is the imaginary component. In our example, after completing the square, we found \( (y + 1)^2 = -1 \). Taking the square root gives us \( y + 1 = \pm i \), where \( i \) represents \( \sqrt{-1} \). This results in the complex solutions \( y = -1 + i \) and \( y = -1 - i \).
Understanding complex numbers is crucial for solving equations that do not have real solutions.

Square Roots

The square root operation helps reverse the effect of squaring a number. When you have \( x^2 = a \), taking the square root of both sides gives \( x = \pm \sqrt{a} \).

It's important to remember the plus/minus (\( \pm \)) when solving equations since any number squared (be it positive or negative) results in a positive value initially.

• If \( a \) is positive, \( \sqrt{a} \) is a real number solution.
• If \( a \) is negative, the square root of \( a \) will involve complex numbers.

In our original equation, after completing the square we ended with \( (y + 1)^2 = -1 \). By taking the square root, we found \( y + 1 = \pm \sqrt{-1} \), resulting in \( y + 1 = \pm i \). Hence, understanding square roots is vital for handling both real and complex solutions.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form \( ax^2 + bx + c = 0 \). These equations can have up to two solutions, which may be real or complex numbers.
There are various methods to solve quadratic equations, including:

• Factoring
• Using the quadratic formula
• Completing the square

Each method has its advantages based on the context of the equation.

Completing the square, for instance, is particularly useful when the quadratic equation doesn't easily factor or when exploring complex solutions.
In this exercise, we dealt with a quadratic \( y^2 + 2y + 2 = 0 \), applied the completing the square method, and ended up with solutions involving complex numbers. Solving quadratic equations is a fundamental skill in algebra, and mastering different methods enhances problem-solving flexibility.