How Do You Convert a Quadratic Equation to Vertex Form?

Converting a quadratic equation to vertex form involves completing the square. This process transforms the equation from its standard form (ax² + bx + c) to vertex form (a(x - h)² + k), where (h, k) represents the vertex of the parabola.

Steps to Convert to Vertex Form

Here's a detailed breakdown of the steps:

1. Start with the quadratic equation in standard form: ax² + bx + c

2. Factor out 'a' from the x² and x terms: a(x² + (b/a)x) + c

3. Complete the square:

   • Take half of the coefficient of the x term inside the parentheses (which is b/2a).
   • Square it: (b/2a)² = b²/4a²
   • Add and subtract this value inside the parentheses: a(x² + (b/a)x + b²/4a² - b²/4a²) + c

4. Rewrite the expression inside the parentheses as a perfect square trinomial: a((x + b/2a)² - b²/4a²) + c

5. Distribute 'a' to the second term inside the parentheses: a(x + b/2a)² - a(b²/4a²) + c

6. Simplify the constant term: a(x + b/2a)² - b²/4a + c

7. Combine the constant terms to get a single constant term (k): a(x + b/2a)² + (c - b²/4a)

8. Now the equation is in vertex form: a(x - h)² + k, where h = -b/2a and k = c - b²/4a

Example

Let's convert the quadratic equation x² + 6x + 5 to vertex form.

1. Standard form: x² + 6x + 5 (a = 1, b = 6, c = 5)

2. Factor out 'a' (which is 1 in this case, so no change): 1(x² + 6x) + 5

3. Complete the square:

   • Half of the coefficient of the x term: 6/2 = 3
   • Square it: 3² = 9
   • Add and subtract inside the parentheses: (x² + 6x + 9 - 9) + 5

4. Rewrite as a perfect square trinomial: (x + 3)² - 9 + 5

5. Simplify: (x + 3)² - 4

Therefore, the vertex form of the quadratic equation x² + 6x + 5 is (x + 3)² - 4. The vertex of the parabola is (-3, -4).

Summary

Converting a quadratic equation to vertex form involves completing the square. This method allows you to easily identify the vertex of the parabola represented by the equation.