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Chapter 8: Problem 39

Use the square root property to solve each equation. \((3 x-1)^{2}=7\)

Short Answer

Expert verified
The solutions are \( x = \frac{\sqrt{7} + 1}{3} \) and \( x = \frac{-\sqrt{7} + 1}{3} \).

Step by step solution

01

Take the Square Root of Both Sides

To start, apply the square root to both sides of the equation: \( \sqrt{(3x - 1)^{2}} = \sqrt{7} \)
02

Simplify the Square Root

When taking the square root of a squared term, the square root and the square cancel each other out. This leaves: \(3x - 1 = \pm \sqrt{7} \)
03

Solve for x

We now have two equations to solve: \(3x - 1 = \sqrt{7} \) \(3x - 1 = -\sqrt{7} \)
04

Add 1 to Both Sides

For each equation from Step 3, add 1 to both sides to isolate the term with x: \(3x = \sqrt{7} + 1 \) \(3x = -\sqrt{7} + 1 \)
05

Divide Both Sides by 3

Finally, divide each equation by 3 to solve for x: \(x = \frac{\sqrt{7} + 1}{3} \) \(x = \frac{-\sqrt{7} + 1}{3} \)

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Key Concepts

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

Square Root Property
Understanding the square root property helps simplify equations, especially quadratics. The square root property states that if you have an equation in the form \(x^2 = k\), then \(x = \pm \sqrt{k}\). This means there will be two solutions: one positive and one negative. Applying this to our exercise, we have \( (3x - 1)^2 = 7 \), which fits the form. By taking the square root of both sides, we can significantly reduce the complexity of the equation.
Isolate Variable
To solve for a variable, it's important to isolate it on one side of the equation. In algebra, this means using operations like addition, subtraction, multiplication, and division. For our equation, after applying the square root property, we need to solve for \(x\) by first isolating the term \(3x - 1\). This is done by getting rid of any constants: Add or subtract terms to both sides of the equation. In our case, adding one to both sides simplifies the expression, making it easier to handle.
Simplify Square Root
When working with square roots, simplifying is a crucial step. After taking the square root of both sides of the equation, you often deal with expressions like \( \sqrt{7} \). Simplifying square roots involves evaluating them if they are perfect squares or leaving them in their simplest form if they are not. For instance, the square root of 7 is already in its simplest form. Hence, the solutions to our equation contain \( \sqrt{7} \) as an essential part.
Quadratic Equation Solutions
Quadratic equations often have two solutions. Once the square root property has been applied and the variable isolated, you'll end up with two linear equations to solve: one for the positive root and one for the negative. For \((3x - 1)^2 = 7\), this gives us \(3x - 1 = \sqrt{7}\) and \(3x - 1 = -\sqrt{7}\). Solving these separately yields two values for \(x\). This approach ensures you've covered all possible solutions for the quadratic equation.

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Most popular questions from this chapter

Solve each problem. When appropriate, round answers to the nearest tenth. Manuel is planting a vegetable garden in the shape of a right triangle. The longer leg is \(3 \mathrm{ft}\) longer than the shorter leg, and the hypotenuse is \(3 \mathrm{ft}\) longer than the longer leg. Find the lengths of the three sides of the garden.

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