/*! 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 69 Solve the equations in Exercises... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 69

Solve the equations in Exercises 53-72 using the quadratic formula. \(6 x^{2}+6 x+1=0\)

Short Answer

Expert verified
The roots of the equation are \(x = \frac{-6 + \sqrt{12}}{12}\) and \(x = \frac{-6 - \sqrt{12}}{12}\).

Step by step solution

01

Identify the coefficients

From the equation \(6 x^{2}+6 x+1=0\), we identify \(a=6, b=6\), and \(c=1\). These will be used in our quadratic formula.
02

Substitute the coefficients into the quadratic formula

Substitute the coefficients into the quadratic formula. This gives: \(x = \frac{-6 \pm \sqrt{(6)^{2} - 4*6*1}}{2*6}\).
03

Simplify the quadratic formula

We can further simplify the equation to: \(x = \frac{-6 \pm \sqrt{36 - 24}}{12}\) which simplifies to: \(x = \frac{-6 \pm \sqrt{12}}{12}\).
04

Calculate the roots

This gives us two solutions, which are the roots of our equation: \(x = \frac{-6 + \sqrt{12}}{12}\) and \(x = \frac{-6 - \sqrt{12}}{12}\). Simplify these to get the final answer.

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Key Concepts

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

Quadratic Formula
The quadratic formula is an essential tool in solving quadratic equations of the form \( ax^2 + bx + c = 0 \). Its primary purpose is to find the values of \( x \) (the roots) that satisfy this equation. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] It provides a straightforward way to find solutions by substituting the coefficients \( a \), \( b \), and \( c \) from the given quadratic equation. This formula is especially useful when other methods like factoring are not suitable.
Coefficients in Quadratics
When dealing with quadratic equations, understanding coefficients is crucial. Coefficients are the numerical factors in terms such as \( ax^2 \), \( bx \) and \( c \) in the equation \( ax^2 + bx + c = 0 \). In every quadratic, you have:
  • \( a \): The coefficient of \( x^2 \)
  • \( b \): The coefficient of \( x \)
  • \( c \): The constant term
These coefficients are vital as they help determine the curve's shape, position, and, most importantly, the roots through the quadratic formula. Identifying \( a \), \( b \), and \( c \) correctly helps ensure accurate substitutions into any solutions methods like the quadratic formula.
Simplifying Radicals
Simplifying radicals is an important step when using the quadratic formula. It involves reducing the square root expression \( \sqrt{b^2 - 4ac} \) to its simplest form. For example, \( \sqrt{12} \) can be simplified. Look for perfect squares within the radical, which in this case, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} \). Since \( \sqrt{4} = 2 \), this simplifies to \( 2\sqrt{3} \). Simplifying radicals helps in getting more precise roots and makes the final calculation steps smoother.
Solving Quadratic Equations
Solving quadratic equations means finding the values of \( x \) that make the equation true. The quadratic formula is one effective method to solve these equations. Let's break the solution into steps:
  • Identify the coefficients: From the quadratic \( ax^2 + bx + c = 0 \), determine \( a \), \( b \), and \( c \).
  • Substitute into the formula: Insert these into the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Simplify the expression: Break down the radical and fractions as much as possible to simplify the calculation.
  • Calculate the roots: You'll usually arrive at two possible solutions, depicted by the \( \pm \) in the formula. Simplifying each step ensures accurate results.
By following these steps, you can systematically solve any quadratic equation using the quadratic formula.

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Most popular questions from this chapter

Use FOIL to find the products in Exercises 1-8. \((2 x-9)(7 x-4)\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(3 x^{2}=5 x-1\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}-3 x=18\)

Describe ways in which solving a linear inequality is different than solving a linear equation.

Solve each equation by the method of your choice. \(\frac{x^{2}}{3}-\frac{x}{2}-\frac{3}{2}=0\)
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