Math Problem Statement

change 2x^2+4x+6 to vertex form

Solution

To convert the quadratic equation 2x2+4x+62x^2 + 4x + 6 into vertex form, we use the process called completing the square. Here's the step-by-step process:

Given equation:

y=2x2+4x+6y = 2x^2 + 4x + 6

Step 1: Factor out the coefficient of x2x^2

Factor out the 2 from the first two terms (since the coefficient of x2x^2 is 2):

y=2(x2+2x)+6y = 2(x^2 + 2x) + 6

Step 2: Complete the square

Now, focus on the expression inside the parentheses: x2+2xx^2 + 2x. To complete the square, take half of the coefficient of xx (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and 12=11^2 = 1.

So, we add and subtract 1 inside the parentheses to maintain equality:

y=2(x2+2x+11)+6y = 2(x^2 + 2x + 1 - 1) + 6 y=2((x+1)21)+6y = 2((x + 1)^2 - 1) + 6

Step 3: Simplify

Now distribute the 2 and simplify the expression:

y=2(x+1)22+6y = 2(x + 1)^2 - 2 + 6 y=2(x+1)2+4y = 2(x + 1)^2 + 4

Final result (vertex form):

The vertex form of the equation is:

y=2(x+1)2+4y = 2(x + 1)^2 + 4

In this form, the vertex is at (1,4)(-1, 4).


Would you like more details or have any questions?

Here are five related questions for further practice:

  1. Convert 3x2+6x+53x^2 + 6x + 5 to vertex form.
  2. Find the vertex of the equation y=x2+4x7y = -x^2 + 4x - 7.
  3. What is the axis of symmetry for y=2(x+1)2+4y = 2(x + 1)^2 + 4?
  4. Convert 2x2+8x3-2x^2 + 8x - 3 to vertex form.
  5. 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