Math Problem Statement
Solution
You are looking to convert the quadratic function into the form , which is called the vertex form. To do this, we need to complete the square. Let me walk you through the steps:
Step 1: Factor out the coefficient of from the first two terms.
We have:
Step 2: Complete the square inside the parentheses.
To complete the square, take half of the coefficient of (which is 7), square it, and add and subtract it inside the parentheses:
Step 3: Simplify the expression.
Now, distribute the factor of 3:
Convert -36 to have a denominator of 4:
Final Answer:
The quadratic function in vertex form is:
This is the desired form , where:
- ,
- , and
- .
Would you like more details or have any questions?
Here are five related questions you could explore:
- How do you derive the vertex from the vertex form of a quadratic equation?
- What is the importance of completing the square in solving quadratic equations?
- How does the coefficient affect the shape of the parabola?
- What is the axis of symmetry for the given function?
- How would you find the x-intercepts of this quadratic function?
Tip: When completing the square, always ensure that you maintain balance in the equation by adjusting constants both inside and outside the parentheses correctly.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Completing the Square
Algebra
Formulas
Standard form of a quadratic function: ax^2 + bx + c
Vertex form of a quadratic function: a(x - h)^2 + k
Theorems
Method of Completing the Square
Suitable Grade Level
Grades 9-11
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