using the quadratic formula and completing the squares
The Unit In Summary
You are going to learn how to use the quadratic formula to find the two x intercepts, use the discriminant to figure out the amount of x intercepts there would be, and use the completing the squares method to find the vertex form and the vertex of the parabola.
Using "Completing the squares" method to find the vertex
NOTE: Recall the vertex form y=a(x-h)+k
- You would have a standard form equation to start and you would bracket the first two terms.
- Next you would divide the the terms inside the brackets by it's GCF, then what ever you put the GCF in front of the bracket.
- Then you would divide the second term in the bracket by it's GCF and square the answer
- You would then put your answer inside the bracket as the third term and put it's opposite outside the bracket(after the bracket).
- Then you would multiply the term in the front of the bracket and the term that is right after the bracket
- You would keep the term in front of the bracket the same, but the term after would change to the product of the previous multiplication operation, then you would add/ subtract the product and the term for the variable K
- then you would put your equation in vertex form based on the information given in your work
Example 1: Using the quadratic formula | Quadratic equations | Algebra I | Khan Academy
Using the discriminant
Now that you have seen the video on using the quadratic formula, We can use a part of that formula to find the discriminant. Using the discriminant is to find out how many solutions/ there will be. The formula for the discriminant is the terms under the square root in the quadratic formula.(d=b^2-4ac)
- If the discriminant is equal to zero, then there is 1 solution
- If the discriminant is equal to a number more than 1 then it would have 2 solutions
- If the discriminant is equal to a number less than 1 it will have 0 solutions
- Example: y=6^2-4(3)(-4)
Therefore, there are 2 possible solutions
Example 2: y=1^2-1(1)(1)
Therefore, There is 1 possible solution
Example 3: y=1^2-5(-4)(-3)
Therefore, there is 0 possible solutions
Now you would take everything you learned and apply it to your word problems
NOTE: height is asking for the k
- Maximum height is asking for the vertex
- When the question ask for the maximum height and the time the object would reach that height, you are looking for the vertex
Reflections on Quadratics
During Quadratics, one thing that I have realized is that all the methods are different but the results are usually the same. For example every method can connect you into the vertex form, and from there you would know no what to do. Also realized how much Quadratics related to the world around us through word problems and application questions.