Math

QuestionSolve for xx in the equation 3x+3=35x+3110\frac{3}{x}+3=\frac{3}{5 x}+\frac{31}{10}.

Studdy Solution

STEP 1

Assumptions1. The equation is 3x+3=35x+3110\frac{3}{x}+3=\frac{3}{5x}+\frac{31}{10}

STEP 2

First, we want to eliminate the fractions to simplify the equation. To do this, we can find the least common multiple (LCM) of the denominators.
The denominators are xx, 5x5x, and 1010. The LCM of these is 10x10x.

STEP 3

Multiply each term in the equation by 10x10x to eliminate the fractions.
10x3x+10x3=10x35x+10x311010x \cdot \frac{3}{x} +10x \cdot3 =10x \cdot \frac{3}{5x} +10x \cdot \frac{31}{10}

STEP 4

implify each term.
30x+30x2=6x+31x30x +30x^2 =6x +31x

STEP 5

Rearrange the equation to bring all terms to one side. This will give us a quadratic equation.
30x2+24x31x=030x^2 +24x -31x =0

STEP 6

implify the equation.
30x2+24x31=030x^2 +24x -31 =0

STEP 7

This is a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c =0. We can solve it using the quadratic formula, which is x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}.

STEP 8

Substitute a=30a =30, b=24b =24, and c=31c = -31 into the quadratic formula.
x=24±24243031230x = \frac{-24 \pm \sqrt{24^2 -4 \cdot30 \cdot -31}}{2 \cdot30}

STEP 9

implify under the square root.
x=24±576+37260x = \frac{-24 \pm \sqrt{576 +372}}{60}

STEP 10

Calculate the value under the square root.
x=24±429660x = \frac{-24 \pm \sqrt{4296}}{60}

STEP 11

Calculate the square root.
x=24±6660x = \frac{-24 \pm66}{60}

STEP 12

olve for the two possible values of xx.
x=24+6660orx=246660x = \frac{-24 +66}{60} \quad or \quad x = \frac{-24 -66}{60}

STEP 13

implify to find the two possible values of xx.
x=4260orx=9060x = \frac{42}{60} \quad or \quad x = \frac{-90}{60}

STEP 14

implify the fractions to find the final solutions.
x=710orx=32x = \frac{7}{10} \quad or \quad x = -\frac{3}{2}So the solutions to the equation are x=710x = \frac{7}{10} and x=32x = -\frac{3}{2}.

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