SUMMARY
The equation 2x^2 − 12x + p = q(x − r)^2 + 10 requires finding the values of p, q, and r that satisfy it for all x. The correct values are q = 2, r = 3, and p = 28, as established by comparing corresponding coefficients of the polynomial. The transformation of 2x^2 − 12x into the vertex form 2(x-3)^2 is crucial for accurate coefficient comparison. Understanding this method is essential for solving polynomial equations effectively.
PREREQUISITES
- Understanding of polynomial equations and their forms
- Familiarity with the concept of corresponding coefficients
- Knowledge of vertex form of a quadratic equation
- Basic algebraic manipulation skills
NEXT STEPS
- Study the method of completing the square in quadratic equations
- Learn about polynomial transformations and their implications
- Explore the concept of coefficient comparison in algebra
- Practice solving quadratic equations in different forms
USEFUL FOR
Students studying algebra, educators teaching polynomial equations, and anyone looking to enhance their understanding of quadratic transformations and coefficient analysis.