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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, we're going to learn about the Completing the Square method. This method helps us turn a quadratic equation into a perfect square trinomial, making it easier to find the roots. Can anyone tell me what a quadratic equation looks like?
Is it in the form ax² + bx + c = 0?
Exactly! Now, let’s start with the first step of our method: rearranging the equation. What do we do first?
We move the constant to the other side.
Right! Good job! Now let’s take an example.
Now that we have our equation rearranged, the next step is to check the coefficient of x². What should it be?
It should equal one, right? If it’s not, we have to divide.
Correct! That's a crucial step. After that, we need to add and subtract the square of half the coefficient of x. Can someone explain why we do that?
To create a perfect square trinomial!
Exactly! Now, let’s see how we can write it in the form of (x + p)² = q. Can someone provide an example?
After rearranging into a perfect square, the last thing we do is take the square root of both sides. Who remembers the next steps?
We set the two equations equal to each other after taking the square root!
And then solve for x, right?
Great! It’s important to consider both the positive and negative roots. Let’s also summarize the whole process at the end. What are the main steps we just discussed to complete the square?
Rearrange, make the coefficient of x² equal to one, add/subtract the square, and then solve!
A method for solving quadratic equations by rearranging them into a perfect square trinomial.
The form resulting from completing the square, allowing for easier solving.
The systematic approach involves rearranging, adjusting coefficients, and simplifying.
Completing the square is really quite fair, rearrange, make one, nothing to scare.
Once in a math land, a square needed help. It wanted to ensure it stood tall and would not melt. So, a wise number came and rearranged its friends, creating a perfect pair to make sure it never ends.
R-M-PS: Rearrange, Make one, Perfect Square.
SOP: Square, One, Plus (to remember the steps of Completing the Square).
Term
Completing the Square
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Term
Perfect Square Trinomial
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Term
Quadratic Equation
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An equation of the form ax² + bx + c = 0 where a, b, and c are real coefficients and a ≠ 0.
A trinomial that can be expressed as the square of a binomial, (x + p)².
A numerical factor in a term of an algebraic expression.
The expression b² - 4ac which helps determine the nature of the roots.
The Completing the Square method is a technique used to solve quadratic equations by transforming them into perfect square trinomial forms.
This section outlines the Completing the Square method as a systematic way to solve quadratic equations by rearranging and manipulating the equation into a form that can be easily solved. Key steps include isolating the constant term, ensuring the leading coefficient of the squared term is one, and forming a perfect square trinomial.
The Completing the Square method allows us to solve quadratic equations by transforming them into the form of a perfect square trinomial. The section outlines key steps involved in this process:
In this section, students work through example problems that demonstrate these steps, ultimately gaining a solid understanding of how to apply this method effectively.