## Learning Goals

Throughout this unit, you will learn how to:
• Find the x-intercepts using the quadratic formula
• using the AOS value to find the y- intercept
• completing the square (finding the MAX or MIN value
• solve word problems using the quadratic formula/ completing the square

## Summary

Some key points to remember are:

• That the value of a gives you the shape and direction of opening of the parabola
• The value of c is y-intercept of the parabola
• You can rewrite a quadratic relation of the form y= ax² + bx+c in the form y=a(x-h)+k by completing the square
• The values (h,k) in vertex form represent the maximum and minimum point of the parabola
• The vertex is a minimum point when a>0 and a maximum point when a<0.

The quadratic formula is used when a certain quadratic cannot be factored. This formula was developed by completing the square and solving the quadratic ax² + bx + c = 0. That is also the standard form equation. The formula gives us the solutions or roots of the quadratic equation.
This formula is modeled from the standard form equation. All you must do to find the x-intercepts, zeroes or solutions, is to substitute the a, b, and c values into the equation. Once you plug those values in, solve for x.

## Example

Find the x-intercepts, the vertex, and the equation of the axis of symmetry of the quadratic relation y= - 5x² + 8x - 3. Sketch the parabola.

*The first thing you should do I write down everything you know just by looking at the equation.*

a= -5 b= 8 c= -3

*Now substitute the values into the equations.*

Now that you have your two x-intercepts, you can plug these into the AOS equation to find the Axis of Symmetry or in other words, your x-value for the vertex.

Once you find the AOS value, substitute that into the original standard form equation to solve for the y-intercept, as shown below.

To recap, we have found the 2 x-intercepts, the axis of symmetry, and the vertex of this parabola. Use this information to draw a brief sketch of your parabola.

## Word Problem using the Quadratic Formula

The path of a basketball after it is thrown from a height of 1.5m above the ground is given by the equation h = -0.25d² + 2d +1.5, where h is the height , in metres, and d is the horizontal distance, in metres.

a) How far has the ball travelled horizontally, to the nearest tenth of a metre, when it lands on the ground?

*Keep in mind that there can never be a negative distance, so you must use the positive solution.*

b) Find the horizontal distance when the basketball is at a height of 4.5m above the ground.
The ball is at a height of 4.5m off the ground at the horizontal distances of 2m and 6m.

## The Discriminant

The discriminant can be used to determine how many solutions an equation will have. The formula for finding the discriminant is D= b²- 4ac. The discriminant can be used to determine how many solutions an equation will have. The formula for finding the discriminant is D= b²- 4ac. You must substitute the values from your original equation to solve for the discriminant. If the discriminant is more than 0 that means that there will be 2 solutions. If the discriminant is 0 then there will be 1 solution. Finally, if the discriminant is less than 0 there will be no solution (in other words, no x-intercepts).

Example:

D=3²-4(2)(-5)

D=9-4(-10)

D=9+40

D=49

This means that there 2 solutions.

D=2²-4(1)(1)

D=4-4

D=0

This means there 1 solutions.

D=2²-4(5)(2)

D=4-4(10)

D=4-40

D=36

This means there are no solutions.

## Completing the Square

The process of completing the square involves changing the first two terms of a quadratic relation of the form y= ax² + + c into a perfect square while maintaining the balance of the original relation.

STANDARD FORM ---------> VERTEX FORM

This method is used to find the vertex form of the standard form equation. It is also really useful for finding the maximum or minimum value of an equation. To complete the squares you must first make sure that the value of a is 1. If it is not, then you must factor it out of the equation. The next step is to divide the value of b by 2 and then square it. Add this value beside b. Then you must place brackets around your equation while leaving c on the outside. Next, take the number you got from dividing b by 2 and squaring it, and subtract it (or add it depending on the situation) from c. this is complete, simply your equation.

Example: Rewrite y= x² + 8x + 5 in the form y= a(x-h)² + k.

y= x² + 8x + 5

y= (x²+ 8x) + 5

b=8

8/2= (4)² =16

y= (x²+ 8x +16 -16) +5

y= (x²+ 8x +16) -16 + 5

y= (x + 4)² - 11

By looking at the vertex form equation, you can easily find the vertex ( -4, -11) which can help you graph your parabola. This vertex tells you the MINIMUM value of the parabola.

For more examples, watch the video below.

## Word Problems using Completing the Squares

The path of a ball is modelled by the equation y= -x² + 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?

*Do NOT forget to write your therefore statement at the end.*