By: Guneet Jaswal
Unit 3: Standard Form
What is Quadratics?
Standard Form Equation: y=ax^2+bx+c
Summary of Standard Form
How to solve equations using the quadratic formula:
- Determine the a,b and c values in the equation
- Plug these values into the equation
- Solve the equation to find x.
Example: Use the quadratic formula to solve the equations below.
- If the discriminant is greater than 0 than there are 2 solutions.
- If the discriminant is smaller than 0 than there are no solutions.
- If the discriminant is 0 than there is 1 solution.
Completing the Square
y=ax^2+bx+c -> y=a(x-h)^2+k
Completing the square can be used to find a maximum or minimum point or to solve a maximum revenue word problem.
How to complete the square:
- Put brackets around the first two terms in the equation. If the a value is greater than one take it out of the brackets and divide it out of the rest of the terms in the brackets.
- Next, divide the b value in the equation by 2.
- Then, square that value.
- Plug the value you get in the previous step into the equation. This value goes in the brackets after the second term. The number must first be added and then subtracted.
- Then, take the negative value that was added in the brackets outside of them. If there is an a value outside of the brackets, make sure you multiply this value with the a value.
- Find the perfect square of the equation in brackets and simplify the values outside of the brackets.
- Now you have your equation in vertex form and you have completed the square.
Example: Rewrite the equation in the form y=a(x-h)^2+k by completing the square.
Word Problem Using the Quadratic Formula
Here is an example of a word problem using the quadratic formula:
The length of a rectangle is 16cm greater than its width. The area is 35m^2. Find the dimensions of the rectangle to the nearest hundreth of a metre.
Solution to the word problem:
Word Problem Using Completing the Square
The path of a ball is modeled by the equation y=-x^2+2x+3, where x is the horizontal distance, in metres, from a fence and y is the height, in metres above the ground.
a) What is the maximum height of the ball, and at what horizontal distance does it occur?
b) Sketch a graph to represent the path of the ball.
Solutions to the problem: