Rewriting Quadratic Equations: The First Step to Convert Standard Form to Vertex Form

When it comes to working with quadratic equations, there are two main forms you’ll come across: standard form and vertex form. Standard form is written as ax² + bx + c = 0, while vertex form is written as a(x – h)² + k. Converting a quadratic equation from standard form to vertex form can make it easier to identify the vertex of the parabola and solve for other key features like the axis of symmetry and the minimum or maximum value. But before you can convert an equation to vertex form, you need to start by rewriting it in a slightly different format.

The first step to convert a quadratic equation from standard form to vertex form is to complete the square. This involves manipulating the equation so that it takes the form of (x – h)² = 4p(y – k), where (h, k) represents the coordinates of the vertex and p represents the distance between the vertex and the focus of the parabola. To get to this point, you’ll need to follow a few key steps:

Step 1: Make sure the coefficient of x² is equal to 1

If the coefficient of x² is not already equal to 1, divide both sides of the equation by that coefficient.

Step 2: Move the constant term to the other side of the equation

Subtract c from both sides of the equation so that you have ax² + bx = -c

Step 3: Add and subtract (b/2a)² to create a perfect square

Take the coefficient of x, divide it by twice the coefficient of x², square the result, and add that term to both sides of the equation. Then subtract it again to balance the equation.

Step 4: Rewrite the left side of the equation as a perfect square

Rearrange the terms on the left side of the equation so that they are in the form (x + [b/2a])²

Step 5: Simplify the right side of the equation

Simplify the right side of the equation by multiplying out the terms and collecting like terms. The result should be in the form 4p(y – k).

Now that you’ve completed the square, you can easily convert the equation to vertex form by dividing both sides by a and rearranging the terms as necessary. The final equation should be written as a(x – h)² + k = 0, where (h, k) represents the vertex of the parabola.

Rewriting quadratic equations to complete the square is a fundamental skill for working with parabolas, and it’s an important first step in converting equations from standard form to vertex form. Once you’ve mastered this technique, you’ll be able to quickly identify the key features of any parabolic equation and solve problems with ease.