Problem 60 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ y^{2}+8 y+18=0 $$

Step-by-Step Solution

Answer

The solutions are \(y = -4 + i\sqrt{2}\) and \(y = -4 - i\sqrt{2}\).

Step 1: Move the constant term

Start by moving the constant term to the other side of the equation to isolate the quadratic and linear terms. The equation becomes:\[ y^2 + 8y = -18 \]

Step 2: Find the term to complete the square

To complete the square, take the coefficient of the linear term (which is 8), divide it by 2, and then square the result. \[ \left( \frac{8}{2} \right)^2 = 16 \]Add this value inside the square on the left side of the equation. Adjust the right side of the equation accordingly.

Step 3: Add and subtract the square term

Add 16 to both sides of the equation to maintain equality:\[ y^2 + 8y + 16 = -18 + 16 \]

Step 4: Simplify and factor the perfect square

Simplify the right side of the equation:\[ y^2 + 8y + 16 = -2 \]Factor the left side to form a perfect square:\[ (y + 4)^2 = -2 \]

Step 5: Solve the equation

Take the square root of both sides, remembering to include the '±' for the square root:\[ y + 4 = \pm \sqrt{-2} \]This indicates imaginary numbers since the square root of a negative number is not real.

Step 6: Write the final solution

Solve for y by isolating it on one side:\[ y = -4 \pm i\sqrt{2} \]where \(i\) is the imaginary unit.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form \[ ax^2 + bx + c = 0 \]. Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Quadratics often appear in algebra, physics, and engineering problems. Understanding how to solve them is crucial since quadratic equations model many real-world situations.

To solve a quadratic equation, you can use various methods such as factoring, the quadratic formula, or completing the square. Each method offers a different way to find the equation's roots, or the values of \(x\) that make the equation true. In the equation \(y^2 + 8y + 18 = 0\), we use the completing the square method to find the solution.

• Factoring: Useful when the equation can be expressed as a product of its factors.
• Quadratic Formula: An all-encompassing method applicable to any quadratic equation.
• Completing the Square: Involves creating a perfect square trinomial from the quadratic equation.

Recognizing the type of quadratic and applying the appropriate method will help solve these equations effectively.

Imaginary Numbers

Imaginary numbers arise when we take the square root of negative numbers, which is not possible within the realm of real numbers. The imaginary unit, denoted \(i\), is defined by \(i^2 = -1\). Thus, for any negative number \(-a\), its square root can be expressed as \(\sqrt{-a} = i\sqrt{a}\).

In the given solution \(y + 4 = \pm \sqrt{-2}\), taking the square root of \(-2\) indicates the solution involves imaginary numbers. Solving this gives us:

• \(y = -4 + i\sqrt{2}\)
• \(y = -4 - i\sqrt{2}\)

Imaginary numbers extend the number system and play an important role in advanced mathematics, including complex numbers, which consist of a real and an imaginary part.

They have practical applications in engineering, particularly in the analysis of electrical circuits and signal processing, making them integral to both theoretical and applied sciences.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It generally takes the form \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\). Recognizing and factoring perfect square trinomials is an important step in the method of completing the square.

In the equation \(y^2 + 8y = -18\), by determining the term needed to complete the square, we changed the equation to \(y^2 + 8y + 16 = -2\), forming a perfect square trinomial \((y + 4)^2\). This transformation allows the equation to be rewritten in a more solvable form.

• Identify the linear term coefficient, divide it by 2, and square it: \(\left(\frac{8}{2}\right)^2 = 16\).
• Add this square to the equation to form \((y + 4)^2\).

This method provides a systematic approach to solving quadratics when other factoring methods are inefficient or when involving irrational or complex roots. Recognizing these trinomials simplifies solving and reveals the beauty of algebraic structures.