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Chapter 17: Chapter 17

Problem 479

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Solve the equation \(3 \mathrm{x}^{2}-5 \mathrm{x}+2=0 \mathrm{by}\) means of the quadratic formula.

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The solutions to the equation \(3x^2 - 5x + 2 = 0\) using the quadratic formula are \(x = 1\) and \(x = \frac{2}{3}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Identify the coefficients a, b, and c

In the equation, 3x^2 - 5x + 2 = 0, the coefficient of x^2 is a = 3, the coefficient of x is b = -5, and the constant term is c = 2.

Step 2: Plug the coefficients into the quadratic formula

Now we will plug the values of a, b, and c into the quadratic formula as follows: \[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(2)}}{2(3)}\]

Step 3: Simplify the expression inside the square root

First, we will simplify the expression inside the square root. \[\sqrt{(-5)^2 - 4(3)(2)} = \sqrt{25 - 24} = \sqrt{1}\]

Step 4: Simplify the entire fraction

Now that we found the value of the square root, we can simplify the entire fraction. \[x = \frac{5 \pm \sqrt{1}}{6}\]

Step 5: Solve for two possible values of x

There are two possible values of x, one with the plus sign and the other with the minus sign. We need to calculate both values: 1. When we use the plus sign: \[x = \frac{5 + 1}{6} = \frac{6}{6} = 1\] 2. When we use the minus sign: \[x = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}\] So the solutions to the equation 3x^2 - 5x + 2 = 0 are \(x = 1\) and \(x = \frac{2}{3}\).

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