Standard Form to Vertex Form - Quadratic Equations

Summary

The lesson explains how to convert a quadratic expression between standard form and vertex form.

• Standard form: (x^2+6x-5)
• Vertex form: ((x+3)^2-14)

Standard form to vertex form: complete the square

1. Take the coefficient of (x), (6), and halve it: (6/2=3).
2. Square the result: (3^2=9).

3. Add (9) to the quadratic terms to form a perfect-square trinomial. To keep the expression’s value unchanged, subtract (9) from the constant: [ x^2+6x-5=x^2+6x+9-9-5. ]

4. Factor the perfect-square trinomial: [ x^2+6x+9=(x+3)^2. ]

5. Combine the remaining constants: [ (x+3)^2-14. ]

In vertex form, the vertex is ((-3,-14)). The (x)-coordinate can also be checked using (x=-\frac{b}{2a}). For this example, (x=-\frac{6}{2}=-3).

Vertex form to standard form: expand

To convert ((x+3)^2-14) back to standard form:

1. Expand the square: [ (x+3)(x+3)=x^2+3x+3x+9. ]

2. Combine like terms and constants: [ x^2+6x+9-14=x^2+6x-5. ]

The main takeaway is that completing the square converts standard form to vertex form, while expanding the squared binomial converts vertex form to standard form.

Speaker or source

• The Organic Chemistry Tutor — the instructor narrating the lesson.