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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 2: Q. 4 (page 126)

Find the real solutions of the equation:
$6 x^{2} - 5 x + 1 = 0$

Short Answer

Expert verified

The solution of the given equation is:

$x = \frac{1}{2} x = \frac{1}{3}$

Step by step solution

01

Step 1. Given information

The given equation is $6 x^{2} - 5 x + 1 = 0$

02

Step 2. Factorise the equation

Using the factorisation method, we get:

$6 x^{2} - 5 x + 1 = 0 6 x^{2} - 3 x - 2 x + 1 = 0 3 x (2 x - 1) - (2 x - 1) = 0 [t a k i n g o u t 3 x a n d - 1 a s c o m m o n] (2 x - 1) (3 x - 1) = 0$

03

Step 3. Evaluating the value

Using zero product property, we get:

$(2 x - 1) = 0 2 x = 1 x = \frac{1}{2}$

or

$3 x - 1 = 0 3 x = 1 x = \frac{1}{3}$

04

Step 4. Verify the solution

Substituting $x = \frac{1}{2}$ in the given equation, we get:

$6 x^{2} - 5 x + 1 = 0 6 {(\frac{1}{2})}^{2} - 5 (\frac{1}{2}) + 1 = 0 6 脳 \frac{1}{4} - 5 脳 \frac{1}{2} + 1 = 0 \frac{3}{2} - \frac{5}{2} + 1 = 0 \frac{- 2}{2} + 1 = 0 - 1 + 1 = 0 0 = 0$

This is true.

Substituting role="math" localid="1645888581245" $x = \frac{1}{3}$ in the given equation, we get:

$6 x^{2} - 5 x + 1 = 0 6 {(\frac{1}{3})}^{2} - 5 (\frac{1}{3}) + 1 = 0 6 脳 \frac{1}{9} - 5 脳 \frac{1}{3} + 1 = 0 \frac{2}{3} - \frac{5}{3} + 1 = 0 \frac{2 - 5}{3} + 1 = 0 \frac{- 3}{3} + 1 = 0 - 1 + 1 = 0 0 = 0$

This is true.

Thus the solution of the given function is:

role="math" localid="1645888988809"

x = \frac{1}{2} x = \frac{1}{3}

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