Problem 31 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}=0.25 $$

Step-by-Step Solution

Answer

The solutions are $$x = \pm \frac{1}{2}$$.

Step 1: Understand the square root property

The square root property states that if *$$x^{2} = c$$* then $$x = \pm \sqrt{c}$$.

Step 2: Apply the square root property

In this problem, $$x^{2} = 0.25$$. So taking the square root of both sides gives $$x = \pm \sqrt{0.25}$$.

Step 3: Simplify the radical

The square root of 0.25 can be simplified. Since *$$0.25 = \left(\frac{1}{2}\right)^{2}$$* , the square root is *$$\frac{1}{2}$$* . Therefore, *$$\sqrt{0.25} = \frac{1}{2}$$*.

Step 4: State the final solution

So, the solutions to $$x^{2} = 0.25$$ are $$x = \pm \frac{1}{2}$$.

Key Concepts

Understanding the Square Root Property

The square root property is a handy tool when solving quadratic equations, especially those in the form \(x^{2} = c\). This property tells us that if you have an equation where a variable squared equals a constant, you can find the variable by taking the square root of the constant. Mathematically, this is expressed as \(x^{2} = c \Rightarrow x = \pm \sqrt{c}\).
This means for any non-zero constant value \(c\), \(x\) has two possible solutions: the positive and negative square roots of \(c\). Don't forget to include the \(\pm \) sign (which stands for plus or minus). It's crucial as it accounts for both directions of the squared term. For instance, in our problem where \(x^{2} = 0.25\), since 0.25 is the same as \(\left(\frac{1}{2}\right)^{2}\), the possible solutions are \(x = \pm\frac{1}{2}\).

Simplifying Radicals

Simplifying radicals helps make solutions more understandable. Radicals are expressions that contain roots, like square roots. One common simplification is finding the square root of a fraction. For example, \(\sqrt{0.25}\) simplifies to \(\frac{1}{2}\).
Here's the step-by-step on how to simplify radicals:

• First, factor the number under the square root.
• Next, identify and extract any perfect squares.
• Finally, simplify what remains under the root.

For our problem, we identified that \(0.25\) can be written as \(\left(\frac{1}{2}\right)^{2}\). Then, applying the square root, we get \(\sqrt{0.25} = \frac{1}{2}\). Just remember, keeping your radicals simplified makes solving and understanding quadratic equations much easier!

Solutions to Quadratic Equations

When solving quadratic equations, like \(x^{2} = 0.25\), we often end up with two solutions, thanks to the symmetry of the parabola described by quadratic functions. Using the square root property, you identify and include both positive and negative roots.
Here’s the key idea: *when you take the square root of both sides of the equation*, you must account for both \(\sqrt{0.25}\) and \(-\sqrt{0.25}\). This gives us \(x = \pm\frac{1}{2}\).
In general, the solutions to a quadratic equation provide the points where the parabola (graph of the quadratic function) intersects the x-axis. So, for \(x^{2} = 0.25\), the solutions are \(x = \frac{1}{2}\) and \( x = -\frac{1}{2}\). This tells us that the parabola touches the x-axis at \(\frac{1}{2}\) and \(-\frac{1}{2}\). Therefore, it’s important to always include both the positive and negative solutions unless the context specifically directs otherwise.