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Chapter 11: Problem 71

Solve using the square root property. Simplify all radicals. $$ (4 x-1)^{2}-48=0 $$

Short Answer

Expert verified
The solutions are \(\tilde x = \frac{1 + 4\sqrt{3}}{4} , x = \frac{1 - 4\sqrt{3} \4}

Step by step solution

01

Isolate the squared term

First, isolate the squared term \( (4x - 1)^2 \) by adding 48 to both sides of the equation: \[ (4x - 1)^2 - 48 + 48 = 0 + 48 \] which simplifies to \[ (4x - 1)^2 = 48 \]
02

Apply the square root property

Take the square root of both sides to eliminate the square: \[ \sqrt{(4x - 1)^2} = \sqrt{48} \] which gives \[ 4x - 1 = \pm\sqrt{48} \]
03

Simplify the radical

Simplify \sqrt{48} by breaking it into prime factors: \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \] So the equation becomes \[ 4x - 1 = \pm 4\sqrt{3} \]
04

Solve for x

Separate the equation into two cases due to \( \pm \): \[ 4x - 1 = 4\sqrt{3} \] \[ 4x - 1 = -4\sqrt{3} \] Solve each equation by adding 1 to both sides and dividing by 4: \[4x = 4 \sqrt{3} + 1 \] \[x = \sqrt{3} + \frac{1}{4} \] \[4x = -4 \sqrt{3} + 1 \] \[x = -\frac{\frac{1}{4}}{4} \]
05

Write the final solution

Combine the solutions from both cases to get the final answer: \[ x = \frac{1 + 4\frac{\ both\frac{}{to the each}} \]

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Key Concepts

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

square root property
To solve a quadratic equation using the square root property, you start by isolating the squared term. Once isolated, take the square root of both sides of the equation to eliminate the square.
This step reveals two possible solutions due to the property of squares: For example, if you have \( y^2 = k \), then \( y = \pm \sqrt{k} \).
This means there are always two values satisfying the equation: one positive and one negative.
isolate the variable
Isolating the variable is crucial when solving equations. In quadratic equations, this often means isolating the squared term first.
In our exercise, we isolated \( (4x - 1)^2 \) by adding 48 to both sides, transforming the equation to \( (4x - 1)^2 = 48 \). This step clears the path to apply the square root property, allowing us to eventually isolate x itself.
simplifying radicals
Simplifying radicals is an essential skill in solving quadratic equations. When simplifying \( \sqrt{48} \), break it down into its prime factors:
\[\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \].
This transformation makes it easier to work with the equation and find solutions. It's important to recognize and use properties of square roots to ensure radical expressions are in their simplest form.
solving quadratic equations
Solving quadratic equations requires understanding various techniques. In our example, we use the following steps:
  • Isolate the squared term.
  • Apply the square root property.
  • Simplify the resulting radicals.
  • Isolate the variable again.
This method can be very efficient for equations that are already set up conveniently for the square root property. Responsibilities in breaking the method down into these clear, manageable steps helps make the solution process more intuitive.

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Most popular questions from this chapter

In each problem, find the following. (a) A function \(R(x)\) that describes the total revenue received (b) The graph of the function from part ( \(a\) ) (c) The number of unsold seats that will produce the maximum revenue (d) The maximum revenue A charter bus charges a fare of \(\$ 48\) per person, plus \(\$ 2\) per person for each unsold seat on the bus. The bus has 42 seats. Let \(x\) represent the number of unsold seats.

Use the quadratic formula to solve each equation. (All solutions for these equations are non real complex numbers.) $$ z(2 z+3)=-2 $$

Solve for \(x .\) Assume that a and \(b\) represent positive real numbers. \(x^{2}-a^{2}-36=0\)

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ 3 x^{2}=5 x+2 $$

William Froude was a 19th century naval architect who used the following expression, known as the Froude number, in shipbuilding. $$ \frac{v^{2}}{g \ell} $$ This expression was also used by R. McNeill Alexander in his research on dinosaurs. (Data from "How Dinosaurs Ran," Scientific American.) Use this expression to find the value of \(v\) (in meters per second), given \(g=9.8 \mathrm{~m}\) per sec \(^{2}\) (Round to the nearest tenth.) Triceratops: \(\ell=2.8\) Froude number \(=0.16\)
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