July 7, 2024

Deriving a vertex form from a standard form of a quadratic equation

To change a quadratic equation from a standardized form to a vertex form is quite easy to derive. You can certainly do so if you understand what a perfect square is. In this article, we would show you all the ideal concepts regarding vertex form and vertex formula. Let us start with a technique called “Completing the Square”. Using this technique, we can change a quadratic equation into a perfect square and you would get to know that it can be easily factorized. So, let us check how it is done.

Theories regarding a perfect square

Let us start with some examples over the application of the technique called “Completing the Square”. The main goal of CTS or Completing the Square is to take any quadratic equation that is not a perfect square and change it into a squared one without even changing the value. We have a few theories regarding this which is listed below.

Theory #1: Squares can be easily factorized

Any quadratic equation that is a square can be easily factored. Let us take an example:

x2 – 16x + 64 is a square which can again be denoted as (x – 8)2.

Theory #2: Finding the pattern within squares

From the above quadratic equation, we can say that it has a pattern since the leading co-efficient is a perfect square. Thus we can say that squaring half of b is always equivalent to c. So, from the above example we can say that half of b is equal to –(16/2) = -8. Now if we square -8, we get (-8)2 = 64.

Theory #3: Retaining the Value 

Let us take an example of an equation, y=5x – 9. We can add or subtract values in the equation without changing the original value of the equation. For instance, we can add 3 to each part of the equation.

Thus we can write, y + 3 = 5x – 9 +3. However, it is not an appropriate purpose to add in this equation, but mathematically, it does not change the value of the equation. Similarly, we can add and subtract the same values from one part of the equation simultaneously. Using the above example, we can say, 

y = 5x – 9 + 3 – 3. Here we are adding and subtracting 3 within one part of the equation without changing the value of the equation.

Finding the vertex and the vertex form

Let us take another example of an equation for instance.

y = x2 + 8x – 2

This equation cannot be factorized and apparently it is not a perfect square either. As we have said before, to be a perfect square you should square the half of b to get c. But within this equation it is not the same. So, what would be the value of c to make this a perfect square?

c should have to be 16 to make the equation a perfect square. Let us add and subtract 16 within one part of the equation. So the equation looks like, y = x2 + 8x + 16 – 2 – 16. Thus, we get a perfect square, x2 + 8x + 16, with some extra values. 

Let us factor the perfect square and combine the extra values which would lead to:

y = (x + 4)2 – 18.

This is actually the vertex form of the original equation, y = x2 + 8x – 2 and the vertex is (-4, -18).

Thus to summarize, for changing a quadratic equation to vertex form, we need to change it into a perfect square with few extra values.  Eventually, we use the half of b and then square it. After that we add and subtracted the squared value within one part of the equation. Lastly, we factorize the perfect square and combine the extra values.

For further details regarding vertex formula, book a session with Cuemath for online math classes.

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