Math Problem Statement
Which equation is y = (x + 3)2 + (x + 4)2 rewritten in vertex form? y = 2 (x + seven-halves) squared minus one-fourth y = 2 (x + seven-halves) squared minus one-half y = 2(x + 7)2 – 73 y = (x + 7)2 – 24
Solution
We are given the equation:
First, let's expand each term on the right-hand side.
- Expand :
- Expand :
Now, add these two expanded terms together:
Simplifying:
Next, let's convert this equation into vertex form. Vertex form is:
We first factor out the 2 from the quadratic and linear terms:
Now, complete the square on . To complete the square, take half of 7 (which is ) and square it:
Add and subtract inside the parentheses:
Simplify the expression inside the parentheses:
Now distribute the 2:
Convert 25 into a fraction:
Finally, simplify:
Thus, the equation in vertex form is:
However, none of the answer choices match exactly. The closest match is:
Would you like further details or have any questions?
Related Questions:
- How do you complete the square on a quadratic equation?
- What is the significance of vertex form in quadratic functions?
- How do you find the vertex of a quadratic equation in standard form?
- Can you explain why we factor out the 2 when completing the square?
- How does the transformation of change when moving to ?
Tip: Always double-check your arithmetic when completing the square, especially with fractions!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Vertex Form
Completing the Square
Formulas
Quadratic equation standard form: ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Completing the square: (b/2)^2
Theorems
Completing the Square Theorem
Vertex Form of Quadratic Equations
Suitable Grade Level
Grades 9-11
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