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Chapter 1: Problem 60

Find the real or imaginary solutions to each equation by using the quadratic formula. $$9 x^{2}-6 x+4=0$$

Short Answer

Expert verified
The solutions are \[x = \frac{1}{3} + \frac{i\sqrt{3}}{3}\] and \[x = \frac{1}{3} - \frac{i\sqrt{3}}{3}\].

Step by step solution

01

Identify coefficients

In the quadratic equation \[ax^2 + bx + c = 0\], identify the coefficients: \[a = 9\], \[b = -6\], and \[c = 4\].
02

Write down the quadratic formula

The quadratic formula is given by \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\].
03

Substitute coefficients into the formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \[x = \frac{{-(-6) \pm \sqrt{{(-6)^2 - 4(9)(4)}}}}{2(9)}\].
04

Simplify inside the square root

Calculate the discriminant: \[(-6)^2 - 4(9)(4) = 36 - 144 = -108\].
05

Evaluate the square root

Since the discriminant is negative (-108), we use imaginary numbers: \[\sqrt{-108} = \sqrt{-1 \times 108} = \sqrt{-1} \times \sqrt{108} = i \times \sqrt{108}\]. Further simplify: \[\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}\]. Thus, \[\sqrt{-108} = 6i\sqrt{3}\].
06

Substitute back into the quadratic formula

Substitute the simplified square root back into the formula: \[x = \frac{6 \pm 6i\sqrt{3}}{18}\].
07

Simplify the final answer

Simplify the fraction: \[x = \frac{6}{18} \pm \frac{6i\sqrt{3}}{18} = \frac{1}{3} \pm \frac{i\sqrt{3}}{3}\]. So, the solutions are \[x = \frac{1}{3} + \frac{i\sqrt{3}}{3}\] and \[x = \frac{1}{3} - \frac{i\sqrt{3}}{3}\].

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Key Concepts

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

Understanding the Discriminant
The discriminant is a crucial part of the quadratic formula. It is found under the square root in the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \].
To identify the discriminant, look at the expression \ b^2 - 4ac \.
The value of the discriminant helps you determine the nature of the roots:
  • If the discriminant is greater than 0, the quadratic equation has two distinct real solutions.
  • If it equals 0, there is one real solution (a repeated root).
  • If the discriminant is less than 0, the solutions are complex or imaginary numbers.
Working with Imaginary Numbers
When the discriminant is negative, the square root of a negative number involves imaginary numbers. An imaginary number is the square root of -1, denoted by \ i \.
For example, \sqrt{-108} \ is simplified by factoring out \sqrt{-1} \, which equals \ i \. Hence, \[ \sqrt{-108} = i\sqrt{108} = 6i\sqrt{3} \].
Imaginary numbers are helpful in solving equations where no real number solutions exist, expanding our ability to work with all types of quadratic equations.
Solving Quadratic Equations with the Quadratic Formula
The quadratic formula, \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \], is a powerful tool to find the roots of any quadratic equation.
Here鈥檚 a quick guide to solving a quadratic equation using this formula:
  • Identify your coefficients \ a \, \ b \, and \ c \.
  • Calculate the discriminant \ b^2 - 4ac \.
  • Substitute the values of \ a \, \ b \, and \ c \ into the quadratic formula.
  • Simplify the expression under the square root.
  • Evaluate the roots to find your final solutions.
With practice, using the quadratic formula can become a straightforward process for tackling any quadratic equation.

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Most popular questions from this chapter

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