Problem 88 - Algebra and Trigonometry

Question

Solve equation by the method of your choice. $$ 2 x^{2}=250 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5\sqrt{5}\) and \(x = -5\sqrt{5}\)

Step 1: Isolate the \(x^{2}\) term.

Start by isolating the \(x^{2}\) term on one side of the equation. Divide both sides of the equation by the coefficient of \(x^{2}\), which in this case is 2: \[x^{2}= \frac{250}{2} = 125\]

Step 2: Take the square root of both sides.

Next, take the square root of both sides of the equation to solve for \(x\). Since taking the square root of a number produces both a positive and a negative result, there will be two solutions for \(x\): \[x= ±\sqrt{125}\]

Step 3: Simplify the square root.

Finally, simplify the square root if possible. The square root of 125 can be simplified to \(5\sqrt{5}\) (since 125 is equal to 25*5, and 25 is a perfect square), resulting in: \[x= ±5\sqrt{5}\]

Key Concepts

Isolating the Variable

The first step in solving a quadratic equation is to isolate the variable, which in this case is \( x^2 \). This means you want \( x^2 \) alone on one side of the equation. To do this, you'll need to get rid of any coefficients or additional numbers on the same side as \( x^2 \).
Here's how to do it effectively:

• Identify the term that includes the variable you want to isolate. In this equation, it is \( 2x^2 \).
• Look for any coefficients or numbers that are multiplying the variable term. Here, the coefficient is 2.
• To isolate \( x^2 \), divide both sides of the equation by the coefficient, which is 2.
  This looks like: \[x^2 = \frac{250}{2}\]
• Simplify the result to get \( x^2 = 125 \).

By isolating \( x^2 \), you make it easier to solve for \( x \) later in the problem.

Taking Square Roots

Once you have isolated \( x^2 \), the next step is to eliminate the square by taking the square root of both sides of the equation. This step will allow you to find the value(s) of \( x \).
While taking square roots, remember:

• The square root of \( x^2 \) gives you the absolute value of \( x \), which is \( |x| \). Therefore, the result includes both positive and negative solutions.
• Write the equation as \( x = \pm \sqrt{125} \). The symbol \( \pm \) signifies that \( x \) can be either positive or negative.
• Recognize that finding square roots will always yield two possible answers for \( x \) when dealing with real numbers.

By taking the square root of both sides, you can move closer to finding the specific values of \( x \).

Simplifying Square Roots

The final step in solving the equation involves simplifying the square root if possible. Simplifying square roots helps you express the solution in its simplest form.
Here's a step-by-step approach to simplifying \( \sqrt{125} \):

• Start by factoring 125 into its prime factors. 125 can be expressed as \( 5^3 \) (since 125 = 5 * 5 * 5).
• Recognize and utilize the property that allows you to take out pairs of numbers from under the square root. A pair of 5s can be taken out as a single 5.
• Write the simplification as \( \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \).

Thus, the complete solutions for \( x \) become \( x = \pm 5\sqrt{5} \). Simplifying square roots makes the solution more elegant and easier to understand.