Math Problem Statement

Convert -x^2+2x-6 into vertex form

Solution

To convert the quadratic equation x2+2x6-x^2 + 2x - 6 into vertex form, we can complete the square. The vertex form of a quadratic equation is given by:

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

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

Step 1: Factor out the coefficient of x2x^2

Start by factoring out 1-1 from the x2x^2 and xx terms.

y=(x22x)6y = -(x^2 - 2x) - 6

Step 2: Complete the square

Take the coefficient of xx (which is 2-2), divide it by 2, and square it.

(22)2=1\left(\frac{-2}{2}\right)^2 = 1

Add and subtract 1 inside the parentheses to complete the square.

y=(x22x+11)6y = -(x^2 - 2x + 1 - 1) - 6

Step 3: Simplify the expression

Group the perfect square trinomial and simplify.

y=[(x1)21]6y = -[(x - 1)^2 - 1] - 6

Distribute the 1-1.

y=(x1)2+16y = -(x - 1)^2 + 1 - 6

y=(x1)25y = -(x - 1)^2 - 5

Final Answer

The vertex form of x2+2x6-x^2 + 2x - 6 is:

y=(x1)25y = -(x - 1)^2 - 5

The vertex is (1,5)(1, -5).

Do you want any details or have any questions? Here are some related problems for further practice:

  1. Convert x2+6x+8x^2 + 6x + 8 into vertex form.
  2. Find the vertex form of 2x24x+12x^2 - 4x + 1.
  3. Convert x24x+7x^2 - 4x + 7 into vertex form.
  4. Find the vertex form of 3x2+12x9-3x^2 + 12x - 9.
  5. Write 4x28x+54x^2 - 8x + 5 in vertex form.

Tip: Always remember to factor out the coefficient of x2x^2 before completing the square.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Completing the Square

Formulas

Vertex form of a quadratic equation: y = a(x - h)^2 + k
Completing the square

Theorems

The process of completing the square to find the vertex form

Suitable Grade Level

Grades 9-11