/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Solve. \(\sqrt{16 x}=x+3\)... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 7

Solve. \(\sqrt{16 x}=x+3\)

Short Answer

Expert verified
The solutions are \(x=1\) and \(x=9\).

Step by step solution

01

Square Both Sides

To eliminate the square root, square both sides of the equation. This gives us: \[(\sqrt{16x})^2 = (x+3)^2\] which simplifies to \[16x = x^2 + 6x + 9\].
02

Rearrange to Form a Quadratic Equation

Rearrange the equation from Step 1 to form a standard quadratic equation. Move all terms to one side of the equation:\[x^2 + 6x + 9 - 16x = 0\] which simplifies to \[x^2 - 10x + 9 = 0\].
03

Factor the Quadratic Equation

Next, factor the quadratic equation \(x^2 - 10x + 9 = 0\). Look for two numbers that multiply to 9 and add up to -10. These numbers are -1 and -9. Thus, the equation factors to:\[(x-1)(x-9) = 0\].
04

Solve for x

Now that the equation is factored, set each factor equal to zero and solve for \(x\): \[x-1=0 \quad \text{or} \quad x-9=0\] which gives the solutions \(x=1\) or \(x=9\).
05

Check for Extraneous Solutions

Because we squared both sides of the original equation, we must check our solutions in the original equation \(\sqrt{16x} = x+3\) to ensure they are not extraneous. - For \(x=1\), \(\sqrt{16 \times 1} = 1+3\) simplifies to \(4=4\), which is true.- For \(x=9\), \(\sqrt{16 \times 9} = 9+3\) simplifies to \(12=12\), which is also true. Both solutions are valid.

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Key Concepts

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

Factoring Quadratic Equations
Factoring quadratic equations is a key technique in solving complex equations. A quadratic equation typically has the form \(ax^2 + bx + c = 0\). The goal of factoring is to express it as a product of two binomials:
  • Look for two numbers that multiply to \(ac\) and add up to \(b\).
  • Use these numbers to break down the middle term \(bx\).
  • Factor by grouping to achieve the binomial factors.
For instance, if you have the equation \(x^2 - 10x + 9 = 0\), the numbers \(-1\) and \(-9\) multiply to \(9\) and add up to \(-10\). Factoring gives \((x-1)(x-9)=0\). A proper understanding of factoring helps in solving the equation efficiently.
Solving Equations
Once a quadratic equation is factored, solving it becomes straightforward. Each factor set to zero provides the solutions. For the factored equation \((x-1)(x-9)=0\):
  • Setting \(x-1=0\) yields \(x=1\).
  • Setting \(x-9=0\) results in \(x=9\).
These are known as the roots or solutions of the quadratic equation. Solving these equations involves understanding that each solution can satisfy the original form, but verifying them is crucial due to possible extraneous solutions.
Solving quadratic equations through factoring provides a simple method to find solutions without requiring more complex calculations like using the quadratic formula.
Extraneous Solutions
Extraneous solutions are those which appear to solve the equation but do not satisfy the original form. They can arise in problems where both sides of an equation are squared, as was done with \(\sqrt{16x} = x+3\). Squaring introduces potential errors:
  • Always substitute solutions back into the original equation to verify.
  • Check logical consistency 鈥� does it hold true when plugged back in?
When \(x=1\) is substituted back, \(\sqrt{16 \times 1} = 1+3\) confirms the truth \(4=4\). Similarly, for \(x=9\), \(\sqrt{16 \times 9} = 9+3\) verifies true as well \(12=12\). Thus, checking for extraneous solutions prevents incorrect conclusions, ensuring only valid solutions are reported.

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Most popular questions from this chapter

Find the maximum or minimum value of each function. Approximate to two decimal places. Methane is a gas produced by landfills, natural gas systems, and coal mining that contributes to the greenhouse effect and global warming. Projected methane emissions in the United States can be modeled by the quadratic function $$ f(x)=-0.072 x^{2}+1.93 x+173.9 $$ where \(f(x)\) is the amount of methane produced in million metric tons and \(x\) is the number of years after 2000 . (Source: Based on data from the U.S. Environmental Protection Agency, \(2000-2020\) ) (IMAGE CANNOT COPY) A. According to this model, what will U.S. emissions of methane be in \(2009 ?\) (Round to 2 decimal places.) B. Will this function have a maximum or a minimum? How can you tell? C. In what year will methane emissions in the United States be at their maximum/minimum? Round to the nearest whole year. D. What is the level of methane emissions for that year? (Use your rounded answer from part c.) (Round this answer to 2 decimals places.)

Solve. See the Concept Check in this section. Which description of \(f(x)=-213(x-0.1)^{2}+3.6\) is correct? $$ \begin{array}{|l|l|} \hline \text { Graph Opens } & {\text { Vertex }} \\ \hline \text { a. upward } & {(0.1,3.6)} \\ \hline \text { b. upward } & {(-213,3.6)} \\ \hline \text { c. downward } & {(0.1,3.6)} \\ \hline \text { d. downward } & {(-0.1,3.6)} \\ \hline \end{array} $$

Notice that the shape of the temperature graph is similar to the curve drawn. In fact, this graph can be modeled by the quadratic function \(f(x)=3 x^{2}-18 x+56,\) where \(f(x)\) is the temperature in degrees Fahrenheit and \(x\) is the number of days from Sunday. (This graph is shown in blue.) Use this function to answer Exercises 85 and 86. Show that the product of these solutions is \(\frac{c}{a}\)

Use the quadratic formula and a calculator to approximate each solution to the nearest tenth. $$ 2 x^{2}-6 x+3=0 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ f(x)=x^{2}-2 $$
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