/*! 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 38 Let \(f(x)=x^{2} .\) Find \(x\) ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 38

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=11\).

Short Answer

Expert verified
The solutions are \( x = \sqrt{11} \) and \( x = -\sqrt{11} \).

Step by step solution

01

Understand the given function

The given function is \( f(x) = x^2 \). This is a quadratic function where the output, \( f(x) \), is equal to the square of the input, \( x \).
02

Set the function equal to 11

We need to find \( x \) such that \( f(x) = 11 \). Therefore, we set up the equation: \( x^2 = 11 \).
03

Solve for \( x \) by taking the square root of both sides

To solve for \( x \), we take the square root of both sides: \( x = \pm \sqrt{11} \). This gives us two potential solutions: \( x = \sqrt{11} \) and \( x = -\sqrt{11} \).

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Key Concepts

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

solving quadratic equations
Quadratic equations are fundamental in algebra and come in the general form: \[ ax^2 + bx + c = 0 \]. In our exercise, the quadratic equation is even simpler: \[ x^2 - 11 = 0 \]. We use the principle of isolating the variable: \[ x ^2 = 11 \].
To solve for \( x \), we take the square root of both sides. Remember that taking the square root of a number has two results: a positive and a negative solution. Therefore, in this case: \[ x = \pm \sqrt{11} \]. The solutions are:
  • \( x = \sqrt{11} \)
  • \( x = -\sqrt{11} \)
Solving quadratic equations can help in many areas like physics and engineering. Practice more examples to get familiar with the steps.
functions
A function represents a relationship between two sets of numbers. In the context of the given problem, the function is \[ f(x) = x^2 \]. Here, \( f(x) \) outputs the square of the input value, \( x \).
Understanding functions is key to solving problems where you need to find the input that gives a specific output. Here is the summary:
  • The function is given as \( f(x) = x^2 \).
  • To find \( x \) for a given \( f(x) \), set up the equation.
  • Solve the equation by substituting \( f(x) \) with the specific value.
Functions are used everywhere in math and science, so mastering them is essential.
square roots
Square roots help us find a number that, when multiplied by itself, gives the original number. Taking the square root of both sides of an equation is a common method to solve quadratic equations. In our exercise, we take: \[ x ^2 = 11 \], and solve it by taking the square root: \[ x = \pm \sqrt{11} \].
There are two solutions because both positive and negative numbers squared result in the same positive value. Practicing finding square roots is beneficial:
  • Ensure you understand both positive and negative solutions.
  • Use square roots to solve quadratic equations easily.
Comprehend the concept of square roots well, as it will repeatedly appear in algebra and other mathematical topics.

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Most popular questions from this chapter

Solve. $$ 9 x^{2}-16=0 $$

Find (a) the maximum or minimum value and (b) the \(x\) - and \(y\) -intercepts. Round to the nearest hundredth. $$ f(x)=-18.8 x^{2}+7.92 x+6.18 $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+9 x-25 $$

Review finding intercepts and completing the square (Sections \(3.3,6.2,6.3, \text { and } 11.1)\) Find the \(x\) -intercept and the \(y\) -intercept. $$ 3 x+4 y=8 $$

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. Find an equation with integer coefficients for which \(1-\sqrt{5}\) and \(3+2 i\) are two of the solutions.
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