Problem 63 - Algebra 1

Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$m^{2}=32$$

Step-by-Step Solution

Answer

The solutions to the equation are \(m=4\sqrt{2}\) and \(m=-4\sqrt{2}\).

Step 1: Understanding the Square Root Method

The equation is \(m^{2} = 32\) which clearly looks like a perfect situation for using the square root method. This method involves taking the square root of both sides of the equation to solve for the unknown.

Step 2: Applying the Square Root

To solve for \(m\), apply the square root to both sides of the equation. Remember, whenever we take the square root of a square, we must consider both the positive and negative root. Therefore, \(m = \sqrt{32}\) will yield \(m = \pm \sqrt{32}\).

Step 3: Simplifying the Result

Simplify \( \sqrt{32}\) which results into \( \pm 4\sqrt{2}\). This is because \(32=16*2\), and the square root of 16 is 4.

Key Concepts

Square Roots

One of the most fundamental concepts in solving quadratic equations is the square root. A square root of a number is a value that, when multiplied by itself, gives the original number.
Take the number 9, for instance. The square root of 9 is 3 because when you multiply 3 by itself (3x3), you get 9.
The square root method is particularly useful when dealing with equations of the form \(x^2 = a\), like \(m^2 = 32\).
Here's why:

• The unknown variable is already isolated as a square, making it straightforward to apply the square root to both sides.
• This method reveals both potential solutions, as every positive number has two square roots: one positive and one negative.

In our equation, after taking the square root of both sides, you derive \(m = \pm \sqrt{32}\). The \(\pm\) symbol indicates that both the positive and negative roots are considered.

Quadratic Formula

When faced with any quadratic equation, we often think about the quadratic formula as a powerful tool to find its roots. The quadratic formula is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]This formula works for any equation of the form \(ax^2 + bx + c = 0\), providing a universal method to find both roots regardless of the coefficients.

• The formula is reliable because it takes into account all possible outcomes for the quadratic solutions, even when there are complex or irrational numbers.
• It can be applied whenever the equation appears more complicated or when taking square roots directly is not feasible.
• In our case, however, using the square root method was more straightforward because the equation is already somewhat solved. But, if the equation had additional terms or coefficients, the quadratic formula would have been more effective.

Simplifying Radicals

Simplifying radicals is an essential skill, especially in solving quadratic equations. Radicals are expressions that contain a square root, and simplifying them can help bring clarity and precision to your calculations.
For instance, in our solution \( \pm \sqrt{32}\), we simplify \(\sqrt{32}\):

• First, express 32 as 16 multiplied by 2: \(32 = 16 \cdot 2\).
• Find the square root of 16, which is 4, and keep the 2 under the radical sign.
• This reduces \(\sqrt{32}\) to \(4\sqrt{2}\).

Simplifying radicals doesn't change the value but makes it easier to work with and clearer when presenting your solution. In math, a cleaner and more concise answer, like \(\pm 4\sqrt{2}\), is always preferred.