Rational Equation

Question

What is/are the solution(s) to the rational equation 2x/x2+2x+ 1+3= 7/x+1? Enter your answer as a single number, like this:42. Or as two or more numbers separated by commas, like this:42, 53 Round any decimal answers to two decimal places, if necessary.

Answer 1

2X/X² + 2X + 1 + 3 = 7/X + 1 (GIVEN)

Multiply through by X²

2X + 2X³+ X² + 3X² = 7X + X²

Regroup terms

2X³ + 4X² - X² + 2X - 7X = 0

Add/ Subtract terms

2X³ + 3X² - 5X = 0 , Factor X out

X(2X² + 3X - 5) = 0, X= 0

2X² + 3X - 5 = 0

X² + (3/2)X = 5/2

Complete squares

X² + (3/2)X + (3/2)²= 5/2 + (3/2)²

(X + 3/2)² = 10/4 + 9/4

Find square roots of both sides

X + 3/2 = (19/4)^1/2

^{X = -1.5 + 2.18, -1.5 - 2.18}

^{X = 0.68, -3.68}

Answer 2

The "1+3" is ambiguous. Is the original problem as written below?

2x / (x² + 2x + 1) + 3 = 7 / (x+1)

Strictly speaking,

2x/x2+2x+ 1+3= 7/x+1 should be written as (2x/x²) + 2x + 1 + 3 = (7/x) + 1 since no parentheses are used.

Let's solve the first assumed equation.

If so, the the 3 needs a common denominator on the left.

Once that is done you can combine the left, then cross multiple and solve for the values of x.

2x / (x² + 2x + 1) + 3 =

[2x / (x² + 2x + 1)] + [3 (x² + 2x + 1) / (x² + 2x + 1) ] =

[2x / (x² + 2x + 1)] + [ (3x² + 6x + 3) / (x² + 2x + 1) ] =

[2x / (x² + 2x + 1)] + [ (3x² + 6x + 3) / (x² + 2x + 1) ] =

(3x² + 8x + 3) / (x² + 2x + 1)

In hind sight, I should have subtracted 3 from both sides first.

(3x² + 8x + 3) / (x² + 2x + 1) = 7 / (x+1)

then cross multiple to get

(3x² + 8x + 3) (x+1) = 7 (x² + 2x + 1)

(3x² + 8x + 3) (x+1) = 7x² + 14x + 7

(3x³ + 8x² + 3x) + (3x² + 8x + 3) = 7x² + 14x + 7

3x³ + 11x² + 11x + 3 = 7x² + 14x + 7

3x³ + 4x² - 3x - 4 = 0

Solve for this equation. It wil have up to 3 unique roots/solutions.

Graphing this function shows three roots: x = -1, 1, and -1.33 or (-4/3)

Answer 3

Once we apply the common denominator, we deal with only the numerators.

2x + 3x² + 6x + 3 = 7x + 7

3x² + x - 4 = 0

(3x + 4)(x - 1)

3x + 4 = 0

3x = -4

x = -4/3

x - 1 = 0

x = 1

-1.33, 1