# Standard Form

### y = ax² + bx + c

## Learning Goals

- How to find the X-Intercepts of a quadratic relation using the Quadratic Formula

## Summary of the Unit

First, we'll start with standard form...

**"COMPLETING THE SQUARE"**

**This changes a quadratic relation in standard form**

y = 2x² + 12x + 11

**to vertex form**

y = 2 (x+3)² - 7

__Next, we have the Quadratic Formula__

1. X - Intercepts

2. Vertex point

With these 3 points, we can graph our Parabola.

Finally, the last thing to cover is the discriminant. The discriminant is the last part of the quadratic formula, D = ² - 4ac

If D = 0, there is 1 x-intercept.

If D > 0, there are 2 x-intercepts.

If D < 0, there are 0 x-intercepts.

## Quadratic Formula Example

A = 2

B = 12

C = 11

Now in quadratic form, we get x =[ -12 +-12²-4(2)(11)]/2 (2)

x = [ -12 +-12²-4(22)]/4

x = [ -12 +-12²-88]/4

x = [ -12 +-144-88]/4

x = [ -12 +-56]/4

x = [ -12 +- 7.5]/4

= -12 + 7.5 = -4.5

-4.5/4 = -1.125

= -12 - 7.5 = -19.5

-19.5/4 = -4.875

X intercepts are (-1.125,0) and (-4.875,0)

## Completing the Square example

1. Bracket ax² + bx, and factor **NUMBERS ONLY.**

y = 2 (x²+6x) + 11

2. Divide bx by 2 and then square it.

6x/2 = 3

3² = 9

3. Plug in number twice, once negative and once positive into brackets

y = 2(x²+6x+9-9) + 11

4. Bring out the negative term out of the bracket, and factor it by whatever factor is before the bracket.

y = 2(x²+6x+9) + 11 - 18

5. Square Root the first and third values of the bracket and then you're DONE.

y = 2(x²+6x+9) + 11 - 18

y = 2 (x+3)² - 7

## Word Problem ( Completing the Square)

We need to complete the square in order to get vertex form, which will give us the vertex of the rocket.

h = -5(-x²-10x) +1.3

10/2 = 5

5² = 25

h = -5(x²-10x + 25 - 25) +1.3

h = -5(x²-10x + 25) +1.3 +125

h = -5(x²-10x + 25) +126.3

h = -5(10x + ) +126.3

h = -5(x + 5)²+126.3

Vertex is The max height of the rocket is 126.3 m.