Math

QuestionFind all real solutions for the equation x2+x130=0x^{-2}+x^{-1}-30=0. If multiple, list them as x=x=.

Studdy Solution

STEP 1

Assumptions1. We are looking for real number solutions. The equation is x+x130=0x^{-}+x^{-1}-30=0

STEP 2

To make the equation easier to work with, we can substitute yy for x1x^{-1}. This gives us a quadratic equation in terms of yy.
y=x1y = x^{-1}y2+y30=0y^2 + y -30 =0

STEP 3

Now, we can factor the quadratic equation.
(y5)(y+6)=0(y -5)(y +6) =0

STEP 4

Setting each factor equal to zero gives us the solutions for yy.
y=0ory+6=0y - =0 \quad \text{or} \quad y +6 =0

STEP 5

olving these equations gives us y=5y =5 and y=y = -.
y=5ory=y =5 \quad \text{or} \quad y = -

STEP 6

Now, we can substitute x1x^{-1} back in for yy to find the solutions for xx.
x1=5orx1=6x^{-1} =5 \quad \text{or} \quad x^{-1} = -6

STEP 7

olving these equations gives us x=1/5x =1/5 and x=1/6x = -1/6.
x=1/5orx=1/6x =1/5 \quad \text{or} \quad x = -1/6So the solutions to the equation x2+x130=0x^{-2}+x^{-1}-30=0 are x=1/5x =1/5 and x=1/6x = -1/6.

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