Math Problem Statement
Solution
We are given the equation:
We need to find the values of and .
Step 1: Expand the right-hand side
The right-hand side of the equation can be expanded using the identity . Thus, the equation becomes:
Step 2: Compare both sides of the equation
Now, compare the coefficients of , , and the constant terms on both sides of the equation:
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Coefficient of : Both sides have , so no change here.
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Coefficient of : On the left-hand side, the coefficient is 7. On the right-hand side, it is . So, we get the equation:
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Constant term: On the left-hand side, the constant is , and on the right-hand side, it is . So, we get:
Step 3: Solve for
We already know . Substitute this into the equation :
Now, solve for :
Final Answer:
Let me know if you'd like more details or have any questions!
Here are five related questions for further practice:
- How would you solve a similar equation if the coefficients were different?
- What happens if is not a constant but another variable?
- How can you complete the square for a quadratic expression?
- What are the steps to solving an equation involving ?
- How does this process change if the equation involves cube terms instead of squares?
Tip: When expanding binomials like , always remember to use the identity to avoid mistakes.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Binomial Expansion
Equation Comparison
Formulas
(x + a)^2 = x^2 + 2ax + a^2
Quadratic Expansion Formula
Theorems
Binomial Theorem
Quadratic Coefficient Matching
Suitable Grade Level
Grades 9-11