Math Problem Statement
Solution
The equation given is:
We need to find the values of and .
Step 1: Expand
Expanding :
So the equation becomes:
Step 2: Compare the terms
We can now compare the coefficients of both sides of the equation. On the left-hand side, we have:
On the right-hand side, after expanding:
Now, compare the coefficients of the corresponding powers of :
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Coefficient of : On both sides, the coefficient of is 1, so this doesn't give us any new information.
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Coefficient of : The coefficient of on the left side is 7, and on the right side, it is . Therefore, we have the equation:
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Constant term: The constant term on the left side is , and on the right side, it is . So, we have:
Step 3: Solve for
Substitute into :
Final Answer:
The values of and are:
Would you like further explanation on any step or details about solving similar problems?
Here are five related questions to explore:
- How do you expand expressions like ?
- What is the method of comparing coefficients in equations?
- How do you solve for unknowns when dealing with quadratic equations?
- Can you explain the process of expanding binomials?
- How do constant terms influence the solution of an equation?
Tip: When solving equations by comparing coefficients, always align similar powers of to set up simpler equations for unknowns.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Expanding Binomials
Formulas
(x + a)^2 = x^2 + 2ax + a^2
Theorems
Method of Comparing Coefficients
Suitable Grade Level
Grades 9-11