Problem 10 - Elementary Algebra

Question

Solve the following quadratic equations. \(y^{2}+64=0\)

Step-by-Step Solution

Answer

y = 8i, y = -8i

Step 1: Isolate the quadratic term

Subtract 64 from both sides of the equation: y^{2} = -64

Step 2: Solve for y

Take the square root of both sides to solve for y. Remember to include both the positive and negative solutions:y = \pm \sqrt{-64}Notice that \(\sqrt{-64}\) can be simplified using imaginary numbers:

Step 3: Simplify the square root of a negative number

Recall that \(\sqrt{-1} = i\) (where i is the imaginary unit). Then:\(\sqrt{-64} = \sqrt{-1 \cdot 64} = \sqrt{-1} \cdot \sqrt{64} = i \cdot 8 = 8i\)Therefore, the solutions are:y = \pm8i

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that result from taking the square root of a negative number. The imaginary unit is denoted as \(i\), where \(i = \sqrt{-1}\). These numbers are essential when dealing with quadratic equations that do not have real solutions.
For instance, in the given problem, we have \(\sqrt{-64}\). Since the square root of a negative number isn't a real number, we use the imaginary unit \(i\):
\[\sqrt{-64} = \sqrt{-1 \cdot 64} = \sqrt{-1} \cdot \sqrt{64} = i \cdot 8 = 8i\]
This allows us to represent solutions for equations that otherwise wouldn't have one in real numbers.

• Imaginary numbers broaden the scope of quadratic equations.
• They help solve equations where traditional methods fail.

Solving Quadratic Equations

To solve quadratic equations, follow these general steps:
1. **Isolate the Quadratic Term**: Get the quadratic term by itself on one side of the equation.
2. **Take the Square Root**: Apply the square root to both sides, remembering to consider both positive and negative roots.
3. **Simplify**: If the square root includes a negative number, convert it to an imaginary number.

Let's use these steps to solve our equation \(y^2+64=0\):
1. **Isolate the Quadratic Term**:
\