Math Problem Statement
Which equation is y = 3(x – 2)2 – (x – 5)2 rewritten in vertex form? y = 3 (x minus seven-halves) squared minus StartFraction 27 Over 4 EndFraction y = 2 (x minus 1) squared minus 11 y = 2 (x minus one-half) squared minus StartFraction 53 Over 4 EndFraction y = 2 (x minus one-half) squared minus StartFraction 27 Over 2 EndFraction
Solution
We are given the equation:
We need to simplify this equation to rewrite it in vertex form.
Step 1: Expand each term
First term:
Second term:
Step 2: Combine the expanded terms
Now simplify by combining like terms:
Step 3: Complete the square to rewrite in vertex form
We will complete the square on the quadratic expression .
Factor out the 2 from the first two terms:
Complete the square inside the parentheses:
To complete the square, take half of the coefficient of (which is ), square it, and add it inside the parentheses:
So,
Final answer in vertex form:
This matches the third option:
Would you like further details on this process?
Here are five related questions you could explore:
- How do you complete the square for a quadratic equation?
- How does vertex form differ from standard form?
- What is the vertex of the equation we just derived?
- How do you convert an equation from vertex form back to standard form?
- How does the coefficient of affect the shape of a parabola?
Tip: Completing the square is a powerful technique not only for rewriting equations but also for solving quadratic equations and finding minimum or maximum values of parabolas.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Vertex Form
Completing the Square
Formulas
Vertex form: y = a(x - h)^2 + k
Completing the square
Theorems
The method of completing the square
Properties of parabolas
Suitable Grade Level
Grades 9-11
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