Math Problem Statement
Solution
We are given the equation:
We need to find the values of , , and .
Step 1: Expand the terms on the left-hand side
Expand :
Expand :
Expand :
Step 2: Combine all expanded terms
Now, combine all terms on the left-hand side: Group the like terms (powers of ):
Step 3: Set up equations by comparing coefficients
We now compare the coefficients of , , and the constant terms with the right-hand side .
- Coefficient of :
- Coefficient of :
- Constant term:
Step 4: Solve the system of equations
The system of equations is:
Let me solve this system of equations.
From equation (1), solve for :
Substitute this expression for into equations (2) and (3):
For equation (2):
For equation (3):
Step 5: Solve the two-variable system
We now have the system:
- (Equation 4)
- (Equation 5)
Solve for in terms of from Equation (5):
Substitute this into Equation (4):
Step 6: Solve for and
Substitute into Equation (5):
Now, substitute and into the expression for :
Final Solution:
Would you like a more detailed breakdown or have any questions?
Here are some related questions for practice:
- What happens if we change the equation on the right-hand side?
- How would you approach solving a system with more unknowns?
- Can you find the discriminant of the quadratic expression for ?
- How does the solution change if the coefficients on the right-hand side are different?
- What method can we use to simplify solving this system?
Tip: When solving systems of equations, substitution is often effective when one variable is easy to isolate.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
System of Equations
Polynomial Expansion
Formulas
Expansion of (a + b)(c + d)
System of equations: comparing coefficients of powers of x
Theorems
Polynomial identity theorem
Solving systems of linear equations
Suitable Grade Level
Grades 9-12
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