Math Problem Statement

vertex form of x^2+6x+2

Solution

To rewrite the quadratic equation x2+6x+2x^2 + 6x + 2 in vertex form, we use the method of completing the square.

The vertex form of a quadratic equation is:

y=a(xh)2+ky = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola.

Let's go through the steps:

1. Start with the original equation:

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

2. Isolate the quadratic and linear terms:

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

3. Complete the square:

To complete the square, take the coefficient of xx (which is 6), divide it by 2, and square it:

(62)2=9\left(\frac{6}{2}\right)^2 = 9

Now, add and subtract 9 inside the parentheses:

y=(x2+6x+99)+2y = (x^2 + 6x + 9 - 9) + 2

y=[(x2+6x+9)]9+2y = [(x^2 + 6x + 9)] - 9 + 2

4. Simplify:

The expression inside the parentheses is now a perfect square:

y=(x+3)27y = (x + 3)^2 - 7

Final answer:

The vertex form of the quadratic equation x2+6x+2x^2 + 6x + 2 is:

y=(x+3)27y = (x + 3)^2 - 7

This shows that the vertex is at (3,7)(-3, -7).

Would you like a deeper explanation of any part of this process?

Here are 5 related questions to expand on this topic:

  1. How do you find the vertex from a quadratic equation in standard form?
  2. What does completing the square mean geometrically?
  3. How can the discriminant help in understanding the roots of a quadratic equation?
  4. How do you convert a quadratic equation to factored form?
  5. What is the significance of the axis of symmetry in a parabola?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Completing the Square
Vertex Form

Formulas

Vertex form: y = a(x - h)^2 + k
Completing the square: (b/2)^2

Theorems

Vertex form theorem
Completing the square method

Suitable Grade Level

Grades 8-10