Problem 26 - Algebra A Combined Function

Question

Use the square root property to solve each equation. $$ (y-4)^{2}+18=0 $$

Step-by-Step Solution

Answer

The solutions are \(y = 4 \pm 3\sqrt{2}i\).

Step 1: Identify and Isolate the Square

The goal is to isolate the squared term to apply the square root property. Start by subtracting 18 from both sides of the equation: \((y-4)^2 = -18\).

Step 2: Apply the Square Root Property

To solve for \(y\), apply the square root to both sides of the equation. This involves taking the square root of \((y-4)^2\) and the square root of \(-18\): \(y-4 = \pm \sqrt{-18}\).

Step 3: Simplify the Square Root

Recognize that \(\sqrt{-18}\) can be written as \(\sqrt{18} \cdot i\), as \(i\) is the imaginary unit. Simplify further: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\). So, \(\sqrt{-18} = 3\sqrt{2}i\).

Step 4: Solve for y

Split the equation into its two solutions based on the plus-minus sign and add 4 to both sides: \(y = 4 + 3\sqrt{2}i\) or \(y = 4 - 3\sqrt{2}i\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). They are second-degree polynomials, and one of the common techniques to solve them is using the square root property. This method is particularly useful for equations that can be rewritten in the format \( (x - p)^2 = q \). The square root property allows you to take the square root of both sides, simplifying the equation to \( x - p = \pm \sqrt{q} \).
Sometimes, like in our exercise, you might have to do some preliminary work to isolate the squared term before applying the property.

For the equation \((y-4)^2 + 18 = 0\), subtract 18 from both sides first, giving us \((y-4)^2 = -18\). Now, we can apply the square root property and introduce imaginary numbers to solve for \(y\).

Imaginary Numbers

In mathematics, imaginary numbers are used when the solutions to an equation involve the square root of a negative number. The imaginary unit is represented by \(i\), where \(i^2 = -1\) and \(i = \sqrt{-1}\).

In many real-world problems, you'll encounter situations where numbers under a square root are negative and cannot be solved using real numbers alone. Let's consider the scenario in our exercise: once you isolate the squared term, you find \(y - 4 = \pm \sqrt{-18}\).

• Recognize that \(\sqrt{-18}\) requires an imaginary number for simplification, hence \(\sqrt{-18} = \sqrt{18} \cdot i\).
• Incorporating \(i\) thus allows us to keep progressing with solving the equation.

Embrace imaginary numbers as a valuable tool that extends the number system beyond real numbers, enabling a complete solution to more complex equations.

Simplifying Square Roots

Simplifying square roots is a crucial step in solving quadratic equations, especially when dealing with imaginary numbers. The general process involves expressing the square root of a number in its simplest radical form.

In the given exercise, once you encounter \(\sqrt{-18}\), you'll want to simplify it further. First, break down 18 into a product of perfect squares: \(18 = 9 \times 2\). Thus, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2}\).

• Since \(\sqrt{9} = 3\), the expression can be simplified to \(3\sqrt{2}\).
• Thus, \(\sqrt{-18} = 3\sqrt{2}i\), using the imaginary unit \(i\).

Practicing the simplification of square roots will improve your problem-solving skills. Always look for factors that are perfect squares, as they are your key to simplifying the expression effectively.