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Chapter 17: Problem 507

Find the sum and product of the roots of the equation \(3 x^{2}-2 x+1=0\)

Short Answer

Expert verified
The sum of the roots of the quadratic equation \(3x^2 - 2x + 1 = 0\) is \(\frac{2}{3}\), and the product of the roots is \(\frac{1}{3}\).

Step by step solution

01

Identify the coefficients of the quadratic equation

The given quadratic equation is \(3x^2 - 2x + 1 = 0\). The coefficients are: \(a = 3\) \(b = -2\) \(c = 1\)
02

Use Vieta's formulas to find the sum of the roots

Vieta's formula for the sum of the roots is given as: \(-\frac{b}{a}\). In our case, \(a = 3\) and \(b = -2\). Thus, the sum of the roots is: \[-\frac{b}{a} = -\frac{-2}{3} = \frac{2}{3}\]
03

Use Vieta's formulas to find the product of the roots

Vieta's formula for the product of the roots is given as: \(\frac{c}{a}\). In our case, \(a = 3\) and \(c = 1\). Thus, the product of the roots is: \[\frac{c}{a} = \frac{1}{3}\]
04

Present the final answer

The sum and product of the roots of the quadratic equation \(3x^2 - 2x + 1 = 0\) are as follows: Sum of the roots: \(\frac{2}{3}\) Product of the roots: \(\frac{1}{3}\)

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