Math Problem Statement
change 2x^2+4x+6 to vertex form
Solution
To convert the quadratic equation into vertex form, we use the process called completing the square. Here's the step-by-step process:
Given equation:
Step 1: Factor out the coefficient of
Factor out the 2 from the first two terms (since the coefficient of is 2):
Step 2: Complete the square
Now, focus on the expression inside the parentheses: . To complete the square, take half of the coefficient of (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and .
So, we add and subtract 1 inside the parentheses to maintain equality:
Step 3: Simplify
Now distribute the 2 and simplify the expression:
Final result (vertex form):
The vertex form of the equation is:
In this form, the vertex is at .
Would you like more details or have any questions?
Here are five related questions for further practice:
- Convert to vertex form.
- Find the vertex of the equation .
- What is the axis of symmetry for ?
- Convert to vertex form.
- How does completing the square help in solving quadratic equations?
Tip: The vertex form is useful because it immediately reveals the vertex of the parabola, making it easier to analyze its graph.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Completing the Square
Formulas
Vertex form of a quadratic equation: y = a(x-h)^2 + k
Standard form of a quadratic equation: ax^2 + bx + c
Theorems
Vertex formula
Completing the square method
Suitable Grade Level
Grades 8-10