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Chapter 9: Problem 47

Solve equation by using the square root property. Simplify all radicals. \((5-2 x)^{2}=30\)

Short Answer

Expert verified
x = \frac{5 \pm \sqrt{30}}{2}

Step by step solution

01

- Isolate the squared term

First, observe that the equation is already in a form suitable to apply the square root property. It is \( (5 - 2x)^2 = 30 \).
02

- Apply the square root property

Take the square root of both sides to remove the exponent. This gives us: \[ \sqrt{(5 - 2x)^2} = \sqrt{30} \] Since squaring a number always results in a positive value, we get two possible solutions: \[ 5 - 2x = \pm \sqrt{30} \]
03

- Solve for x

Separate the equation into two parts and solve for \( x \) in each part. \[ 1. 5 - 2x = \sqrt{30} \ \ 2. 5 - 2x = -\sqrt{30} \] \ Next, solve the first equation: \[ 5 - 2x = \sqrt{30} \ \ -2x = \sqrt{30} - 5 \ \ x = \frac{5 - \sqrt{30}}{2} \] \ Now solve the second equation: \[ 5 - 2x = -\sqrt{30} \ \ -2x = -\sqrt{30} - 5 \ \ x = \frac{5 + \sqrt{30}}{2} \]

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Key Concepts

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

Solving Equations
To solve equations, we need to find the value of the variable that makes the equation true. Here, we solve for the variable \(x\). The equation is \((5 - 2x)^2 = 30\).
We use the square root property to solve this type of equation. This property states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).
  • First, isolate the squared term. In this problem, \( (5 - 2x)^2\) is already isolated.
  • Apply the square root on both sides: \((5 - 2x) = \pm \sqrt{30}\).
  • This gives us two possible equations to solve: \(5 - 2x = \sqrt{30}\) and \(5 - 2x = -\sqrt{30}\).
Breaking down equations into smaller parts can help you solve them more easily.
Simplifying Radicals
Simplifying radicals is essential to finding the simplest form of an expression. A radical expression is simplified when there are no perfect square factors other than 1 in the radicand (the number under the radical sign). In these equations, \( \sqrt{30}\) is already in its simplest form because 30 has no perfect square factors except for 1.
When simplifying, follow these steps:
  • Look for any factors in the radicand that are perfect squares.
  • Simplify them by taking the square root.
  • Combine the simplified parts if possible.
In this problem, since \( \sqrt{30}\) can't be simplified further, we use it directly in the solution.
Isolating Variables
Isolating the variable means rearranging the equation so that the variable you are solving for is alone on one side of the equation. This often requires inverse operations.
For the equation \(5 - 2x = \sqrt{30}\), we need to isolate \(x\).
Here's a step-by-step breakdown:
  • First, subtract 5 from both sides: \(-2x = \sqrt{30} - 5\).
  • Divide both sides by -2: \( x = \frac{5 - \sqrt{30}}{2}\).

Then, for the equation \(5 - 2x = -\sqrt{30}\):
  • Subtract 5 from both sides: \(-2x = -\sqrt{30} - 5\).
  • Divide both sides by -2: \( x = \frac{5 + \sqrt{30}}{2}\).
By isolating the variable, you get its value in terms of constants and radicals. Breaking this process into smaller steps can make it easier to follow and understand.

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Most popular questions from this chapter

Answer true or false to each statement. If false, say why. Every real number is a complex number.

Solve equation by using the square root property. Simplify all radicals. \((x-3)^{2}=25\)

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