Problem 32 - Elementary Algebra

Question

Solve using the Square Root Property. \(4 t^{2}=16\)

Step-by-Step Solution

Answer

t = ±2

Step 1: Isolate the squared term

The given equation is \(4t^2 = 16\). Divide both sides by 4 to isolate the squared term.\[t^2 = \frac{16}{4}\]Simplify the division:\[t^2 = 4\]

Step 2: Apply the Square Root Property

To solve for \(t\), take the square root of both sides of the equation. Recall that the square root property gives both the positive and negative solutions.\[t = \pm \sqrt{4}\]

Step 3: Simplify the square root

Simplify the square root:\[t = \pm 2\]

Key Concepts

Solving Quadratic Equations

Quadratic equations are fundamental in algebra and often show up in various math problems. An equation is quadratic if it can be phrased in the standard form \[ax^2 + bx + c = 0\]. In our exercise, the equation was simpler, \[4t^2 = 16\]. To solve any quadratic equation, we can employ various strategies, one of which is the Square Root Property. This method simplifies solving equations where a variable is squared. Understanding how and when to use different techniques for solving quadratic equations is essential for mastering algebra.

Isolating Squared Terms

Isolating the squared term is usually the first step in solving an equation using the Square Root Property. In our example, we started with \[4t^2 = 16\]. The goal here is to separate the \(t^2\) term on one side to simplify the equation. We achieved this by dividing both sides of the equation by 4, giving us \[t^2 = 4\]. This simplified equation is much easier to solve. As a rule of thumb in algebra, always perform the same operation on both sides of the equation to maintain equality.

Positive and Negative Solutions

When using the Square Root Property, it is crucial to remember that squaring is a two-way street. Squaring both positive and negative numbers leads to a positive result. Therefore, applying the square root to \(t^2 = 4\) means we must consider both the positive and negative roots. We write this as \(t = \pm \sqrt{4}\), which simplifies further to \(t = \pm 2\). Always ensure to account for both the positive and negative solutions when dealing with squared terms.