Problem 33 - Elementary Algebra

Question

Solve by extracting the roots. $$ y 2-0.09=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = 0.3 \) and \( y = -0.3 \).

Step 1: Isolate the Square Term

The given equation is \( y^2 - 0.09 = 0 \). First, we should isolate the square term by adding 0.09 to both sides of the equation to get \( y^2 = 0.09 \).

Step 2: Take the Square Root

To solve for \( y \), take the square root of both sides of the equation. This gives us \( y = \pm \sqrt{0.09} \).

Step 3: Simplify the Square Root

Calculate \( \sqrt{0.09} \). Since \( 0.09 \) is the square of 0.3, we find \( \sqrt{0.09} = 0.3 \). Thus, the solutions to the equation are \( y = \pm 0.3 \).

Key Concepts

Solving by Extracting Roots

When it comes to quadratic equations, one efficient method of solving them is **extracting roots**. This technique is applicable when the quadratic is already set up nicely or can be rearranged to allow easy extraction. To extract roots, you first need to isolate the squared term.
In our example, the equation is \( y^2 - 0.09 = 0 \). Notice that the squared term, \( y^2 \), can be isolated by adding 0.09 to both sides of the equation, which results in \( y^2 = 0.09 \).
Now, with the squared term isolated, you proceed by taking the square root of both sides of the equation. This step transforms your equation so you can solve for the variable, \( y \), giving \( y = \pm \sqrt{0.09} \). This pivotal extraction step leads us right to the equation's solutions.

Simplifying Square Roots

Simplifying square roots is crucial in making sure our solutions are as neat as possible. Once you have extracted the root in a quadratic equation, as seen in \( y = \pm \sqrt{0.09} \), it's time to simplify.
For the number 0.09, it's beneficial to realize it's a perfect square because \( 0.09 = (0.3)^2 \). To simplify \( \sqrt{0.09} \), you determine that \( \sqrt{0.09} = 0.3 \), a straightforward squaring.
Recognizing perfect squares helps in simplifying square roots efficiently. It is often helpful to memorize perfect squares and roots of basic numbers. This understanding makes solving equations quicker and more accurate.

Positive and Negative Solutions

Quadratic equations, when solved by extracting roots, often yield **two solutions** due to the nature of squaring and square roots. When you take the square root of a number in a mathematical equation, you must consider both the positive and the negative values.
For example, in our equation, you end up with \( y = \pm 0.3 \). This is because both \( 0.3^2 \) and \((-0.3)^2\) equal 0.09. Hence, it’s essential not to overlook the negative solution.
Emphasizing both positive and negative solutions ensures that you account for all possible scenarios while solving quadratic equations. This is fundamental since real-world problems can often have more than one valid solution. Always remember to reflect both solutions in your final answer to provide a comprehensive solution.