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Chapter 1: Problem 88

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

Short Answer

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

Step by step solution

01

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: \[x^{2}= \frac{250}{2} = 125\]
02

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 \(x\): \[x= 卤\sqrt{125}\]
03

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: \[x= 卤5\sqrt{5}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90. a. What must you get on the final to earn an \(A\) in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is 3000 dollar and it costs 3.00 dollar to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

Find all values of \(x\) satisfying the given conditions. $$ y_{1}=\frac{3}{x-1}, y_{2}=\frac{8}{x}, \text { and } y_{1}+y_{2}=3 $$

How is the quadratic formula derived?

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
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