Math Problem Statement
12
Solution
The given equation is:
Let's solve this equation step by step.
Step 1: Factor the quadratic expression
The quadratic expression in the third fraction is . We can factor this expression:
So the equation becomes:
Step 2: Simplify the first fraction
The first fraction can be simplified by factoring out the 2 from the denominator:
The equation now is:
Step 3: Find a common denominator
The common denominator for all three fractions would be . Notice that , so the common denominator can be expressed as:
Thus, rewrite each term with this common denominator:
Simplifying, we get:
Step 4: Simplify the numerator
Let's expand and simplify the numerator:
Expand :
Combine like terms:
So the equation now becomes:
Step 5: Set the numerator equal to zero
For the equation to be zero, the numerator must be zero:
This is a quadratic equation. We can solve it using the quadratic formula:
where , , and . Plugging in these values:
So the roots are:
Simplify the expressions:
Step 6: Exclude extraneous solutions
We need to exclude any solutions that make the original denominators zero. The denominators were:
These give potential exclusions and . Check these values against the roots:
- is not 4 or -3.
- is not 4 or -3.
So both roots are valid.
Final Answer:
The solutions are:
Would you like more details or have any questions?
Here are some additional questions to consider:
- What is the importance of factoring quadratic expressions in solving equations?
- How does the quadratic formula help in solving any quadratic equation?
- Why do we need to check for extraneous solutions?
- What are the possible implications of having a negative discriminant in the quadratic formula?
- How can you simplify complex fractions involving polynomials?
Tip: When solving quadratic equations, always simplify the expression as much as possible before applying the quadratic formula to avoid unnecessary complexity.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Factoring
Formulas
Quadratic formula
Theorems
-
Suitable Grade Level
High School
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