Problem 21
Question
Solve each equation. For equations with real solutions, support your answers graphically. $$(3 x-1)^{2}=12$$
Step-by-Step Solution
Verified Answer
The solutions are \( x = \frac{1 + 2\sqrt{3}}{3} \) and \( x = \frac{1 - 2\sqrt{3}}{3} \).
1Step 1: Expand the Equation
First, we aim to clear the square by taking the square root of both sides. The equation is \( (3x-1)^2 = 12 \). By taking the square root of both sides, we have \( 3x - 1 = \pm \sqrt{12} \).
2Step 2: Simplify the Square Root
Next, simplify the square root. \( \sqrt{12} \) can be rewritten as \( \sqrt{4 \times 3} = 2\sqrt{3} \), so \( 3x - 1 = \pm 2\sqrt{3} \).
3Step 3: Solve for x
Now, solve the linear equation for \( x \). First, add 1 to both sides to get \( 3x = 1 \pm 2\sqrt{3} \). Then divide by 3 to isolate \( x \), yielding \( x = \frac{1 \pm 2\sqrt{3}}{3} \).
4Step 4: Express the Two Solutions
From \( x = \frac{1 \pm 2\sqrt{3}}{3} \), we get two solutions: \( x_1 = \frac{1 + 2\sqrt{3}}{3} \) and \( x_2 = \frac{1 - 2\sqrt{3}}{3} \).
5Step 5: Graphical Interpretation
To support graphically, plot \( y = (3x-1)^2 \) and \( y = 12 \) on a coordinate plane. The intersections of the parabola \( y = (3x-1)^2 \) with the line \( y = 12 \) occur at the \( x \)-values that solve the equation, corresponding to the solutions \( x_1 \) and \( x_2 \).
Key Concepts
Solving EquationsGraphical SolutionsSquare Roots
Solving Equations
To solve the given quadratic equation \((3x-1)^2=12\), we begin by eliminating the square to make the equation easier to work with. Taking the square root of both sides helps to simplify it. Doing this leads to the equation \(3x - 1 = \pm \sqrt{12}\). This process, called "solving by square roots," is often used for equations where the variable is squared.
When you take the square root, remember to consider both the positive and negative roots. This means that you have two equations to solve: \(3x - 1 = +\sqrt{12}\) and \(3x - 1 = -\sqrt{12}\). Solving these linear equations will help you find the possible values for \(x\).
Once you simplify \(\sqrt{12}\) to \(2\sqrt{3}\), add 1 to both sides, and then divide by 3, you obtain the final answers: \(x_1 = \frac{1 + 2\sqrt{3}}{3}\) and \(x_2 = \frac{1 - 2\sqrt{3}}{3}\). These values represent the solutions to the original equation.
When you take the square root, remember to consider both the positive and negative roots. This means that you have two equations to solve: \(3x - 1 = +\sqrt{12}\) and \(3x - 1 = -\sqrt{12}\). Solving these linear equations will help you find the possible values for \(x\).
Once you simplify \(\sqrt{12}\) to \(2\sqrt{3}\), add 1 to both sides, and then divide by 3, you obtain the final answers: \(x_1 = \frac{1 + 2\sqrt{3}}{3}\) and \(x_2 = \frac{1 - 2\sqrt{3}}{3}\). These values represent the solutions to the original equation.
Graphical Solutions
Graphically representing solutions can be a powerful method to visualize and confirm your calculations. Drawing the problem on a graph helps reveal the nature of the solutions. For the equation \((3x-1)^2 = 12\), plotting the function \(y = (3x-1)^2\) as a parabola along with the horizontal line \(y = 12\) enables us to see where these two graphs intersect.
The points of intersection indicate the solutions' values for \(x\). Sometimes seeing is believing—especially when you can visualize the curve of the parabola intersecting the line \(y=12\) at exactly two distinct points corresponding to \(x_1\) and \(x_2\). By analyzing these points, double-checking through graphing offers reassurance that the algebraic solutions are correct.
Graphical solutions provide a clear visual method for verifying equations with real solutions, offering insight into their nature and ensuring accuracy.
The points of intersection indicate the solutions' values for \(x\). Sometimes seeing is believing—especially when you can visualize the curve of the parabola intersecting the line \(y=12\) at exactly two distinct points corresponding to \(x_1\) and \(x_2\). By analyzing these points, double-checking through graphing offers reassurance that the algebraic solutions are correct.
Graphical solutions provide a clear visual method for verifying equations with real solutions, offering insight into their nature and ensuring accuracy.
Square Roots
Understanding square roots is essential when dealing with equations like \((3x-1)^2 = 12\). The square root process helps undo the squaring of a binomial, making it possible to solve for \(x\). Here, \(\sqrt{12}\) simplifies to \(2\sqrt{3}\), using the rule that allows breaking square roots down into simpler factors.
Square roots can appear daunting, but the trick is to recognize when they can be simplified. For instance, since 12 equals \(4 \times 3\), and \(\sqrt{4} = 2\), we can express \(\sqrt{12}\) as \(2\sqrt{3}\). This simplification makes it easier to work with and shows why both solutions include a part with \(2\sqrt{3}\).
Taking time to understand where square roots come from and how they are used ensures you can confidently tackle similar quadratic equations. Recognizing patterns within numbers often unlocks critical steps necessary for solving more complex problems.
Square roots can appear daunting, but the trick is to recognize when they can be simplified. For instance, since 12 equals \(4 \times 3\), and \(\sqrt{4} = 2\), we can express \(\sqrt{12}\) as \(2\sqrt{3}\). This simplification makes it easier to work with and shows why both solutions include a part with \(2\sqrt{3}\).
Taking time to understand where square roots come from and how they are used ensures you can confidently tackle similar quadratic equations. Recognizing patterns within numbers often unlocks critical steps necessary for solving more complex problems.
Other exercises in this chapter
Problem 21
Write each number in simplest form, without a negative radicand. $$\sqrt{-100}$$
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Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in t
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Give a short written answer. The graphs of \(f(x)=x^{n}\) for \(n=3,5,7, \ldots\) resemble each other. As \(n\) gets larger, what happens to the graph?
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Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{3}+2 x^{2}-17 x-10 ; \quad x+5$$
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