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Solve each rational equation. list excluded values
x+4/x+5=6/x^2+10+25

User Viator
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To solve the given rational equation:

x+4/x+5=6/x^2+10x+25

We will begin by first finding the LCD (Least Common Denominator) which is:

(x+5)(x+5)= (x+5)^2

Multiplying both sides of the equation by the LCD, we get:

(x+4)(x+5)(x+5) = 6(x+5)^2

Expanding both sides of the equation and simplifying further, we get:

x^3 + 14x^2 + 61x + 80 =0

Now, we can use the rational root theorem to identify the possible rational roots of the polynomial equation. The possible rational roots are:

±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40, ±80

Testing these rational roots, we get that x = -4 is a root of the polynomial. Using synthetic division, we can then factor the polynomial as:

(x+4)(x^2 + 10x + 20) = 0

This gives us the solutions:

x = -4, x = -5 + 3sqrt(2)i and x = -5 - 3sqrt(2)i

Therefore, the excluded values are x = -5 and x = -5 ± sqrt(2)i.
User FZs
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