Problem 20 - Algebra A Combined Function

Question

Use the square root property to solve each equation. $$ 3 p^{2}+36=0 $$

Step-by-Step Solution

Answer

The solutions are \( p = 2i\sqrt{3} \) and \( p = -2i\sqrt{3} \).

Step 1: Move Constant to the Other Side

To begin, isolate the term that contains the square by subtracting 36 from both sides of the equation: \[ 3p^2 = -36 \]

Step 2: Divide by the Coefficient of the Square Term

Divide both sides by 3 to solve for \( p^2 \): \[ p^2 = \frac{-36}{3} = -12 \]

Step 3: Apply the Square Root Property

To solve for \( p \), take the square root of both sides. Remember to include the \( \pm \) symbol due to the property of square roots: \[ p = \pm \sqrt{-12} \]

Step 4: Simplify the Square Root

The square root of \( -12 \) involves imaginary numbers because it's negative. Express \( -12 \) as \( 12 \times -1 \), and use \( \sqrt{-1} = i \): \[ p = \pm i\sqrt{12} \]

Step 5: Simplify the Radicand

Simplify \( \sqrt{12} \) as \( \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). Thus, \[ p = \pm 2i\sqrt{3} \]

Key Concepts

Imaginary Numbers

Imaginary numbers come into play when we deal with the square root of a negative number. In mathematics, the unit imaginary number is represented by \( i \), where \( i = \sqrt{-1} \). This means when you take the square root of a negative number, you multiply by \( i \) to indicate it is imaginary.

Why do we use imaginary numbers?

• Imagine you attempt to find the real square root of a negative number, like \( -1 \). In the real number system, there is no real solution, so we introduce \( i \).
• Imaginary numbers allow us to solve equations that were previously impossible, like those that result in negative square roots when solved.

In our example, we have \( \sqrt{-12} \). You split this into \( \sqrt{12} \) and \( \sqrt{-1} \). The \( \sqrt{-1} \) becomes \( i \), thus giving \( i\sqrt{12} \). This step is key in managing the square root of negative numbers, ensuring you correctly identify imaginary numbers in your results.

Solving Quadratic Equations

Solving quadratic equations often involves applying the square root property, which is a method used to find the roots of an equation in the form \( ax^2 + bx + c = 0 \). For the equation \( 3p^2 + 36 = 0 \), we use this property step-by-step.

Steps involved:

• Isolate the square term: Move any constant term away from the term with the square. Here, \(3p^2 = -36 \).
• Divide by the square term coefficient: To simplify, divide by the coefficient of \( p^2 \), resulting in \( p^2 = -12 \).
• Square root both sides: Take the square root, remembering to apply \( \pm \) because both positive and negative roots are solutions to the squared term.

Understanding these steps helps us solve many quadratic equations efficiently, especially those leading to imaginary numbers.

Simplifying Radicals

Simplifying radicals is an essential skill for handling square roots efficiently, especially when they involve imaginary numbers.

We simplify \( \sqrt{12} \) into \( 2\sqrt{3} \) as follows:

• Factorize the number: Break down 12 into its prime factors: \( 12 = 4 \times 3 \).
• Simplify the square: Since \( \sqrt{4} \) is a perfect square (\( 2 \)), extract it from under the square root. Thus, \( \sqrt{12} = 2\sqrt{3} \).

For the equation \( p = \pm i\sqrt{12} \), simplifying the radicand leads to \( p = \pm 2i\sqrt{3} \).

This simplification process is vital to make radicals more manageable and reveal the simplest form of the solution.