Share
Solve for x in the equation x^2 + 3x + 7 = 6 / (x^2 + 3x + 2)
Published on October 31, 2024
Algebra
The equation x^2 + 3x + 7 = 6/(x^2 + 3x + 2) is solved by substituting u = x^2 + 3x, resulting in a quadratic equation in u. Factoring the quadratic equation yields two possible values for u, which are then substituted back into the substitution to find the solutions for x using the quadratic formula. One case results in two real solutions, while the other yields no real solutions.
Question
Answer
Here’s how to solve the equation
-
Substitution: Let
. The equation becomes . -
Clear the fraction: Multiply both sides by
to get . -
Expand and simplify: Expand the left side to get
. Subtract 6 from both sides to get . -
Factor the quadratic: The quadratic factors as
. -
Solve for u: This gives two possible solutions for
: or . -
Substitute back for x:
-
Case 1: u = -1
Using the quadratic formula: -
Case 2: u = -8
Using the quadratic formula:
Since the discriminant is negative, there are no real solutions in this case.
-
Therefore, the solutions for
Final Answer: The final answer is