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Chapter 16: Problem 49

Solve by taking square roots. $$25 x^{2}-64=0$$

Short Answer

Expert verified
The potential solutions to the equation \(25x^{2}-64 = 0\) are either \(x = \frac{8}{5}\) or \(x = -\frac{8}{5}\)

Step by step solution

01

Simplify Equation

First, we'll simplify equation by adding 64 to both sides to isolate the expression with the unknown variable \(x\) on one side. Here's how it works: \(25 x^{2} = 64 \)
02

Divide by 25

Next, we'll divide both sides of the equation by 25 so that \(x^{2}\) remains alone on one side of the equation. This will look like:\[x^{2} = \frac{64}{25}\]
03

Apply Square Root

Now, we want to cancel out the squared part of \(x\). To do this, we'll take the square root of both sides of the equation. Remember, when we take the square root of \(x^{2}\), we must consider the positive and negative root. So, the final solutions will look like:\[x = \sqrt{\frac{64}{25}} = 卤\frac{8}{5}\]

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Key Concepts

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

Solving Equations
Solving equations is like finding the missing piece of a puzzle. It involves figuring out the value of an unknown variable, often represented by a letter like \( x \). In the exercise we're looking at, you start by isolating expressions involving \( x \). Here's a simple way to think about it:
  • First, make the equation neat by getting all terms with the variable on one side. This often involves using basic operations like addition or subtraction. In our case, we added 64 to both sides.

  • Next, isolate the variable to further simplify the problem. This might involve dividing both sides to remove coefficients, just like how we divided by 25 to simplify \(25x^2=64 \).
The goal is to have a simple equation where the variable can be easily determined. Each step helps get closer to unveiling the unknown, making the solution clearer.
Square Roots
Square roots are fascinating because they unravel squares, showcasing the hidden simplicity behind them. In mathematical terms, the square root of a number \( y \) is a number \( x \) such that \( x^2 = y \). In our exercise, once we have \( x^2 = \frac{64}{25} \), we take the square root of both sides to solve for \( x \).
  • The square root operation undoes squaring. When you take the square root of \( x^2 \), you end up with \( x \) or \(-x \).

  • That's why solutions are often written as \( 卤 \), representing both positive and negative solutions.
Square roots remind us that numbers have multiple perspectives - often in opposite directions, like two sides of a coin.
Simplifying Expressions
Simplifying expressions is about making complex problems easier to handle. It's like rearranging words in a sentence so it makes more sense. In the scenario from the exercise, simplifying began by translating \( 25x^2 - 64 = 0 \) into \( x^2 = \frac{64}{25} \) by isolating parts of the equation. To simplify effectively:
  • Combine like terms and perform arithmetic operations to condense the expression.

  • It's vital to simplify fractions too, as in our case with \( x = \sqrt{\frac{64}{25}} \). By breaking down \( \frac{64}{25} \) into \( \frac{8}{5} \) by recognizing perfect squares, the expression feels much tidier.

Simplifying clears away clutter, allowing the core of the problem to stand out, making it more approachable and much easier to solve.

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Most popular questions from this chapter

A tank has two drains. One drain takes 16 min longer to empty the tank than does the second drain. With both drains open, the tank is emptied in 6 min. How long would it take each drain, working alone, to empty the tank?

If it takes one pipe 15 min longer to fill a tank, than it does a second pipe, and the rate of work for the second pipe is represented by \(\frac{1}{1},\) then the rate of work for the first pipe can be represented by ______.

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State the number of real number solutions. $$(x+a)^{2}=0$$

Solve by taking square roots. $$y^{2}+81=0$$
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