/*! 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 54 $$ \text { For Problems 45-68,... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 54

$$ \text { For Problems 45-68, solve each equation. (Objective 2) } $$ $$ 49 x^{2}=36 $$

Short Answer

Expert verified
The solutions are \( x = \frac{6}{7} \) and \( x = -\frac{6}{7} \).

Step by step solution

01

Isolate the Quadratic Term

The given equation is \( 49x^2 = 36 \). First, divide both sides by 49 to isolate the \( x^2 \) term: \[ x^2 = \frac{36}{49}. \]
02

Take the Square Root

To solve for \( x \), take the square root of both sides: \[ x = \pm \sqrt{\frac{36}{49}}. \]
03

Simplify the Square Root

Simplify the expression \( \sqrt{\frac{36}{49}} \): \[ x = \pm \frac{\sqrt{36}}{\sqrt{49}}. \] Since \( \sqrt{36} = 6 \) and \( \sqrt{49} = 7 \), we get \[ x = \pm \frac{6}{7}. \]
04

Write the Final Answer

The solutions to the equation \( 49x^2 = 36 \) are \( x = \frac{6}{7} \) and \( x = -\frac{6}{7} \).

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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 often show up in various math courses and can be recognized by the standard form, \( ax^2 + bx + c = 0 \). In these equations, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Solving quadratic equations involves finding the values of \( x \) that make the equation true.

There are several methods to solve quadratic equations, such as:
  • Factoring
  • Using the quadratic formula
  • Completing the square
  • Square root method
Choosing the right method depends on specific characteristics of the equation.

Here, since the equation is in the form \( 49x^2 = 36 \), it's more straightforward to use the square root method.This technique simplifies the problem-solving process, especially when the equation can be rearranged easily.
Square Root Method
The square root method is a quick and efficient way to solve quadratic equations when the quadratic term is already isolated.This technique is especially helpful when equations are in the form \( ax^2 = c \).

To use the square root method, follow these steps:
  • Isolate the quadratic term: Begin by making sure \( x^2 \) is on one side of the equation.
  • Take square root of both sides: Apply the square root operation to both sides, remembering to include both the positive and negative roots, \( x = \pm \sqrt{c} \).
  • Simplify the result: Break down the square roots individually, if possible.
This method works efficiently when the right-side constant can be expressed as a perfect square. In our example,\( 49x^2 = 36 \) can be simplified using the square root method to provide the solution \( x = \pm \frac{6}{7} \). This approach is direct, limiting complex manipulations and ensuring accuracy.
Simplifying Fractions
Simplifying fractions is a basic yet essential skill in mathematics. It involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors besides 1.

To simplify fractions, let's consider the example \( \frac{36}{49} \):
  • Find the greatest common divisor (GCD) of the numerator and denominator. In this case, it's already simplified, as 36 and 49 have a GCD of 1.
  • Simplify: Divide both the numerator and the denominator by their GCD, if necessary.
After taking square roots, we obtain \( \sqrt{36}/\sqrt{49} \), which simplifies to \( 6/7 \) as both numbers are perfect squares.

This step is crucial in finding the exact value of the solutions to our quadratic equation. By understanding how to simplify fractions, you ensure your solutions are as clear and concise as possible.This makes your answers easier to interpret and apply.

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Most popular questions from this chapter

\(x(x-10)=-16\)

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) A number is one less than three times another num-ber. If the product of the two numbers is 102 , find the numbers.

\(n^{2}-6 n-40=0\)

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 4 x^{2}-20 x=0 $$

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Suppose that a square and a rectangle have equal areas. Furthermore, suppose that the length of the rectangle is twice the length of a side of the square, and the width of the rectangle is 4 centimeters less than the length of a side of the square. Find the dimensions of both figures.
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