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Chapter 7: Problem 11

Solve each equation.. \((6 x-5)^{2}=0\)

Short Answer

Expert verified
Using the square root property, we find that \(6x-5 = 0\). Solving for x, we get \(x = \frac{5}{6}\).

Step by step solution

01

Simplify the Equation

Since \((6x-5)^2 = 0\), to simplify it, we'll need to find the square root of both sides of the equation. \[\sqrt{(6x-5)^2} = \sqrt{0}\]
02

Use the Square Root Property

We know that the square root of a square is equal to the value inside the parenthesis. So, we have: \[6x - 5 = 0\]
03

Solve for x

Now it's time to isolate x and solve the equation. To do this, we'll add 5 to both sides of the equation: \[6x = 5\] Now divide by 6 to get the value of x: \[x = \frac{5}{6}\] So, the solution to the equation \((6x - 5)^2 = 0\) is \(x = \frac{5}{6}\).

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Key Concepts

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

Square Root Property
The square root property is a foundational concept when addressing quadratic equations in forms like \((6x - 5)^2 = 0\). This simply turns the equation into a linear format. To apply this property, take the square root of both sides of the equation.
  • For example, take \((6x - 5)^2\) and find the square root, which results in \(6x - 5\).
  • Do the same for the other side of the equation, such as \(0\), which remains 0 after applying the square root.
This property holds because a square root undoes a square, simplifying the equation's form. Recognizing when to use square root property simplifies handling quadratics to a more manageable state.
Isolate Variable
Isolating the variable is a crucial step in solving any equation, including quadratics. After applying the square root property, you would typically see an equation like \(6x - 5 = 0\). From here, you have a clearer path to isolate \(x\). Here's how you do it:
  • First, balance the equation by adding or subtracting terms. In this case, add 5 to both sides to remove the constant term associated with \(x\).
  • This gives you \(6x = 5\).
The aim here is to 'move' other terms away from the variable until it's by itself. Every operation you perform should be done on both sides of the equation to maintain balance.
  • Once the equation is \(6x = 5\), divide both sides by the coefficient of \(x\), which is 6 in this scenario.
Appearing simple, isolating dictates clearer solutions and step-by-step accuracy to find the unknown.
Simplify Equation
Simplifying an equation involves condensing it to its simplest form, removing complex symbols or grouped terms. In quadratics like \((6x - 5)^2 = 0\), simplifying is key as a first step:
  • This starts by identifying opportunities to condense expression, like recognizing a squared term and using its root.
  • After simplifying using the square root property, you move towards isolating your variable, performing basic arithmetic operations.
Let's break it down further. Arithmetic simplicity complements larger operations like square roots. For example:
  • You first break down \(6x - 5\) to 0. Then add 5 to both sides. This is part of arithmetic simplification aimed at easing the way to isolating \(x\).
Ultimately, simplifying any equation is crucial to not only solving it effectively but also ensuring that each step toward the solution is seamless, clear, and comprehensible.

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