Unit 3 : Standard Form
By : Mrinal Bhavsar
Learning Goals :
- Properly learn how to complete the square.
- Be able to solve expansion problems (application problems).
- Be able to explain how the discriminant defines the number of x - intercepts.
Summary of the Unit
The standard form of a quadratic equation is 0 = ax² + bx + c. The exponent of two is what makes this a quadratic function. The values a, b, and c are known, and the value of a cannot be 0. The value of x is the unknown variable.
How to solve it?
The solutions to the quadratic equation are the roots, or called the zeroes, which are the x-intercepts of the parabola when graphed. There are two ways to solve for this including:
- Completing the Square : This technique is used to find the vertex of the parabola so that it can be graphed. The steps are to factor if necessary, then take the middle term, divide by two, and square it. Then add and subtract the new number to the equation. This will result in the vertex form of the equation, which can then be used to graph the function.
- Using the Quadratic Formula : This technique is used to find the x - intercepts of the parabola (if there are any). To determine the number of x - intercepts you can use the discriminant. The discriminant is the number inside the square root in the quadratic formula. If the discriminant is less than 0, then there are no possible solutions. If the discriminant is greater than 0, then there are two x - intercepts. If the discriminant is 0, then there is only one x - intercept. The image below shows what the quadratic formula is. All you have to do is take the a, b, and c values in the standard form equation, and sub them into the quadratic formula, then solve for x.
Example of solving with the Quadratic Formula
Example of solving by Completing the Square
Word Problem Using the Quadratic Formula
Reflection
I feel like Quadratics was a very fun unit to learn. I learned that completing all homework and studying was the key to success to getting better grades on assessments. The assessment I think I did best on was the Unit Test on Quadratic Relations in Standard Form (examples shown above are from this assessment).
Some connections I have made between the various forms of quadratic relations are how with the standard form you can use greatest common factoring to complete the square, which helps find the vertex, then use the vertex form to graph the relation. Each of the three units in quadratics connect with each other to easily be able to graph the relation.