Problem 3 - Elementary Algebra

Question

Solve the following quadratic equations. \(r^{2}-24=0\)

Step-by-Step Solution

Answer

The solutions are \(r = 2\sqrt{6}\) and \(r = -2\sqrt{6}\).

Step 1 - Rewrite the equation

Start with the given equation: \[ r^{2} - 24 = 0 \] Move the constant term to the other side to isolate the quadratic term: \[ r^{2} = 24 \]

Step 2 - Solve for r

Take the square root of both sides of the equation to solve for r: \[ r = \pm\sqrt{24} \] Simplify the square root if possible: \[ r = \pm 2\sqrt{6} \]

Key Concepts

Quadratic Equations

Understanding quadratic equations is essential in algebra. A quadratic equation is any equation that can be written in the form \[ax^2 + bx + c = 0\]. Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The given problem is a special case of a quadratic equation where \(b\) and \(c\) are zero. Thus, it simplifies to just \[r^2 - 24 = 0\], making the quadratic term \[r^2\] straightforward to handle. One way to solve such equations is by isolating the quadratic term and then taking the square root of both sides.

Square Roots

Taking the square root is a method to find a number which, when multiplied by itself, gives the original number. In the equation \[r^2 - 24 = 0\], after rearranging, we have \[r^2 = 24\]. To isolate \(r\), we take the square root of both sides: \[r = \pm \sqrt{24}\].

Remember, square roots have both positive and negative solutions because \[(-r)^2\] and \[r^2\] yield the same result. Therefore, we write \[r = \pm \sqrt{24}\].

Simplifying Radicals

Simplifying radicals involves breaking them down to their simplest form. In our equation, we have \[\sqrt{24}\]. We can simplify this by finding the prime factors of 24. The prime factors are 2, 2, 2, and 3, thus:

• \[24 = 2 \times 2 \times 2 \times 3\]
• The product contains a pair of 2's: \[2 \times 2 = 4\]

This allows us to simplify \(\sqrt{24}\) to \(2 \sqrt{6}\). So, the final solution can be written as: \[r = \pm 2 \sqrt{6}\]. By simplifying the radical, the equation becomes easier to understand and solve. Simplifying radicals helps avoid misinterpretations and keeps equations neat.