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Chapter 1: Problem 55

Find the real solutions of each equation. $$ x-4 x \sqrt{x}=0 $$

Short Answer

Expert verified
The solutions are \( x = 0 \) and \( x = \frac{1}{16} \).

Step by step solution

01

Factor the Equation

First, factor out the common term from the equation. The given equation is \[ x - 4x \sqrt{x} = 0 \]. Factoring out the common term \( x \), we get \[ x (1 - 4\sqrt{x}) = 0 \].
02

Set Each Factor to Zero

Using the zero-product property, set each factor equal to zero and solve for \( x \). This gives us two equations: 1. \( x = 0 \)2. \( 1 - 4\sqrt{x} = 0 \)
03

Solve the First Equation

The first equation is straightforward: \[ x = 0 \]
04

Solve the Second Equation

For the second equation, rearrange it to solve for \( x \):\[ 1 - 4\sqrt{x} = 0 \]Add \( 4\sqrt{x} \) to both sides:\[ 1 = 4 \sqrt{x} \]Divide both sides by 4:\[ \sqrt{x} = \frac{1}{4} \]Square both sides to eliminate the square root:\[ x = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \]
05

Compile the Solutions

The real solutions to the given equation are \( x = 0 \) and \( x = \frac{1}{16} \).

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Key Concepts

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

Factoring Equations
To solve the equation \[ x - 4x \sqrt{x} = 0 \], we start by factoring. Factoring means breaking down an equation into simpler components or 鈥渇actors鈥� that, when multiplied together, produce the original equation. In this case, we notice that both terms in the equation contain an \( x \). Therefore, we can factor out the \( x \) from the equation, leading to: \[ x(1 - 4\sqrt{x}) = 0 \]. This step simplifies the problem significantly and sets us up for further steps using the zero-product property.
Zero-Product Property
The zero-product property is a core concept in solving equations. It states that if the product of two factors is zero, then at least one of the factors must be zero. After factoring our equation to \[ x(1 - 4\sqrt{x}) = 0 \], we apply this property. This gives us two separate equations to solve: \[ x = 0 \] and \[ 1 - 4\sqrt{x} = 0 \]. By solving each of these factors individually, we can find the solutions to the original equation.
Algebraic Manipulation
Algebraic manipulation involves rearranging an equation to make it easier to solve. For \[ 1 - 4\sqrt{x} = 0 \], we start by isolating the square root term. We add \( 4\sqrt{x} \) to both sides, resulting in: \[ 1 = 4\sqrt{x} \]. Next, we divide both sides by 4 to isolate \( \sqrt{x} \), giving us: \[ \sqrt{x} = \frac{1}{4} \]. These steps are crucial for gradually simplifying and solving the equation.
Solving Equations with Square Roots
Solving equations with square roots involves eliminating the square root by squaring both sides of the equation. For the simplified equation \[ \sqrt{x} = \frac{1}{4} \], we square both sides to get: \[ x = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \]. By squaring, we remove the square root and are left with a straightforward calculation. So, the solutions to our original equation are \( x = 0 \) and \( x = \frac{1}{16} \). Breaking down each part helps to solve complex equations step by step.

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Most popular questions from this chapter

Find \(k\) so that the equation \(x^{2}-k x+4=0\) has a repeated real solution.

The sum of the consecutive integers \(1,2,3, \ldots, n\) is given by the formula \(\frac{1}{2} n(n+1) .\) How many consecutive integers, starting with \(1,\) must be added to get a sum of \(703 ?\)

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the negatives of the real solutions of the equation \(a x^{2}-b x+c=0\). Assume that \(b^{2}-4 a c \geq 0\)

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

Find \(k\) so that the equation \(k x^{2}+x+k=0\) has a repeated real solution.
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