# Quadratics: Standard Form

### By: Shawn Singh

## Success Criteria: Learning Goals

- To
**Complete the Square**in order to find the**maximum/minimum value**of the quadratic and its**vertex** - To use the
**Quadratic Formula**in order to find the**x-intercepts**along with the**vertex**of a quadratic function - To find
**how much x-intercepts a quadratic function has**by finding the**Discriminant** - To
**graph a quadratic function**using the**"Quadratic Formula"**

## Summary

**Standard Form Equation:***y = ax² + bx + c***a**value gives you the**shape and direction of opening**of the quadratic**c**value gives you the**y-intercept**of the quadratic- To get
**x-intercepts**,**SOLVE using the "Quadratic Formula"** **Quadratic Formula:***x =[ -b**on picture*__+__√(b² - 4ac)] / 2a- To identify if the quadratic has a
**maximum or minimum value**,**complete the square to transform standard form equation in vertex form** **Vertex Form Equation:***y = a(x - h)² + k*- To identify how much
**x-intercepts a quadratic has**, use the**discriminant formula** **Discriminant Formula**:*x = b² - 4ac*

## Completing the Square

- Involves
**transforming a standard form equation into vertex form in order to get the vertex and maximum or minimum value of the quadratic**

**steps in order**of

**how to change a standard form equation into vertex form**

**Ex. 1 **

y = x² - 8x + 10

y = (x² - 8x )+ 10 **Step One:** Isolate ax² and bx terms, also (-8/2)² basically (b/2)²

y = (x² - 8x + 16 - 16 ) + 10 **Step Two**: Add and subtract the value from above

y = (x² - 8x + 16) + 10 - 16 **Step 3:** Bring the -16 outside the bracket

y = (√ x² - 8x + √ 16) - 6 **Step 4:** Square root both ax² and c terms)

__y = (x - 4)²__ **Step Five:** Write as a perfect square.

**Ex. 2**

y = 2x² + 20x - 8

y = 2(x² + 10x) - 8 **If there is a coefficient, factor it out (applies to ax² and bx)*

y = 2(x² + 10x + 25 - 25) - 8 **Remember, multiply coefficient with subtracted value (2)(-25)*

y = 2(√ x² + 10x + √ 25) - 50 - 8

__y = 2(x + 5)² - 58__

**Ex. 3**

y = -x² + 10x + 20

y = -(x² - 10x) + 20* *Even if there is a negative sign, factor it out**

y = -(x² - 10x +25 - 25) + 20

y = -(√ x² - 10x + √ 25) + 20 + 25

*y = -(x - 5)² + 45*

**The video below will help further your understanding of Completing the Square**

## The Quadratic Formula

- Using the quadratic formula is a
**direct way of calculating roots (also known as zeroes, the solution or the x-intercepts)** - A quadratic equation
**can have 2, 1 or 0 solutions** - If the
**number under the square root is negative, the quadratic has no solution** *Quadratic Formula:***x =[ -b**__+__√(b² - 4ac)] / 2a

**Ex. 1**

9x² - 24x + 16 = 0

x =[ -b __+__ √(b² - 4ac)] / 2a

x =[ -(-24) __+__ √((-24)² - 4(9)(16))] / 2(9)

x =[ 24 __+__ √(576 - 576)] / 18

x =[ 24 __+__ √0)] / 18

x = (24 + 0) / 18

x = 24 / 18

__x = 1.3 or 4/3__

**(1.3, 0) or (4/3, 0)**

**Ex. 2**

2x² + 4x = -1

2x² + 4x - 1 = 0

x =[ -b __+__ √(b² - 4ac)] / 2a

x =[ -4 __+__ √((4)² - 4(2)(1))] / 2(2)

x =[ -4 __+__ √(16 - 8)] / 4

x =[ -4 __+__ √8] / 4

x = (-4 + 2.8) / 4

x = -1.2/4

__x = -0.3__

x = (-4 - 2.8) / 4

x = -6.8/4

__x = -1.7__

**(-0.3, 0) (-1.7, 0)**

***MY OWN VIDEO BELOW** will help further your understanding of using the Quadratic Formula*

## Graphing Standard Form

**follow these steps:**

**Step One:**Find the x-intercepts using "The Quadratic Formula"**Step 2:**Find the vertex (Use AOS = (r + s)/2 to find x-value and sub in x-value into original equation to find y-value of vertex- Plot the vertex along with the two x-intercepts to form parabola

**Ex. 1**

y = x² + 2x - 3

**Step One: x-intercepts**

x =[ -b __+__ √(b² - 4ac)] / 2a

x =[ -2 __+__ √((2)² - 4(1)(-3))] / 2(1)

x =[ -2 __+__ √(4 + 12)] / 2

x =[ -2 __+__ √(16)] / 2

x = (-2 + 4) / 2

x = 2/2

__x = 1__

x = (-2 - 4) / 2

x = -6/2

__x = -3__

**x- intercepts = (1, 0) (-3, 0)**

**Step Two: Vertex**

AOS = (r + s)/2 = (-3 + 1) / = -2/2 = __-1__

y = x² + 2x - 3

y = (-1)² + 2(-1) - 3

y = 1 - 2 - 3

__y = - 4__

**Vertex (-1, -4)**

**Step 3: Plots x-Intercepts and Vertex on Graph *graph on side***

## The Discriminant

- The Discriminant Formula is used to identify the
**number of x-intercepts in a quadratic function** - Given that
**a, b and c are rational numbers**, they are**needed to determine the Discriminant value.** **Discriminant Formula = b² - 4ac**

**how to determine how much x - intercepts a quadratic function has:**

- D>0 =
**2 Solutions**- If**D****is****greater than 0**the quadratic will have**two x-intercepts.** - D<0 =
**0 Solutions**- If**D is less than 0 the**quadratic will have**no x-intercepts.** - D = 0 =
**1 Solution**- If**D is equal to 0,**the quadratic will have**1 x-intercept.**

**Ex. 1**

y = 2x² + 2x - 3

D = b² - 4ac

= (2)² - 4(2)(-3)

= 4 + 24

__= 28__

**Therefore, since 28 is greater than O, this quadratic will have two x-intercepts. **

**Ex. 2**

y = x² - 2x + 5

D = b² - 4ac

= (-2)² - 4(1)(5)

= 4-20

__= -16__

**Therefore, since -16 is less than 0, this means the quadratic will have no x-intercepts. **

**Ex. 3**

y = x² + 2x + 1

D = b² - 4ac

= (2)² - 4(1)(1)

= 4 - 4

__= 0__

**Therefore, since D = 0, this means the quadratic will have only 1 x-intercept. **

## Word Problem #1 - Complete the Square

**Sherri sells photos of athletes to baseball, basketball and hockey fans after their games. Her regular price is $10 per photo, and she usually sells about 30 photographs. Sherri finds that, for each $0.50 reduction in price, she can sell two more photographs.**

**a) Write an equation to represent Sherri's total sales revenue. **

**[Hint: R = (price)(quantity)]**

Let **x** represent the **number of price decreases.**

__EQUATION:__

*R = (10 - 0.50x)(30 + 2x)*

**b) What price must Sherri charge in order to maximize her revenue. **

R = (10 - 0.50x)(30 + 2x)

= 300 + 20x - 15x - x²

= - x² + 5x + 300

= - (x² - 5x) + 300

= - (x² - 5x + 6.25 - 6.25) + 300

= - (x² - 5x + 6.25) + 300 + 6.25

= - (√x² - 5x + √6.25) + 306.25

__= - (√x² - 2.5)² + 306.25__

**Vertex (2.5, 206.25)**

Price = 10 - 0.50x

= 10 - 0.50(2.5)

= 10 - 1.25

*= 8.75*

Therefore, to maximize her revenue, Sherri must charge **$8.75 per photo. **

## Word Problem #2 - The Quadratic Formula

**The product of two consecutive numbers is 3306. What are the numbers?**

Let **x** represent the **first number.**

Let** x + 1** represent the **second number. **

x(x + 1) = 3306

x² + x = 3306

x² + x - 3306 = 0

x = [-b __+__ √(b² - 4ac)] / 2a

x = [-1 __+__ √((1)² - 4(1)(- 3306))] / 2(1)

x = [-1 __+__ √( 1 + 13224)] / 2

x = [-1 __+__ √13225] / 2

x = (-1 +115) / 2

x = 114 / 2

__x = 57__

x = (-1 -115) / 2

x = -116 / 2

__x = -58__

**x = 57 **

x + 1 = 57 + 1 = **58**

__Check:__ (57)(58) = 3306

**x = -58**

x + 1 = -58 + 1 = **-57**

__Check__: (-58)(-57) = 3306

Therefore, the numbers are **57 and 58 or -57 and -58. **

## Reflection

- Overall the Quadratics Unit has been a fairly interesting unit to learn about where it
**connects to a whole lot of applications throughout the world** - With the exception of a few silly mistakes, I was able to
**exceed throughout this whole unit in all the 3 components and understand as well as apply the key concepts of the quadratics unit**

*Connections: *

**1. Vertex Form Connects to Standard Form**

- Standard Form can be obtained by expanding Vertex Form
*y = a(x – h)² + k -------> y = ax² + bx + c = 0*- Vertex Form can be obtained by "Completing the Square" from Standard Form
*y = ax² + bx + c = 0 -------> y = a(x – h)² + k*

**2. Vertex Form Connects to Graphing**

- Once your quadratic function is put into vertex form you are able to identify the vertex of the function and plot it on your graph
- Once vertex is plotted, use the step pattern (a value included) to continue graphing from both sides

**3. Standard Form Connects to Graphing**

- In a standard form equation, the
*c value gives you the y-intercept*, begin with plotting the y-intercept on the graph - Transforming your Standard Form Equation into Vertex Form by Completing the Square, will give you your vertex as well as your a value
- Once vertex is plotted use the a value as well as the step pattern to continue to plot your quadratic

**4. Factored Form Connects to Graphing**

- In a factored form equation the two x-intercepts as well your a value is easily obtainable by factoring your equation. Plot your two x-intercepts
- Use x-intercepts to find the vertex of the equation
*(x-value - AOS, y - value subbing x into original equation* - Once vertex is plotted use a value along with the step pattern to graph quadratic

**5. Factored Form Connects to Standard Form**

- In order to find the vertex, this two forms involve similar methods involving finding the vertex
__In Factored Form:__Once two x-intercepts are found by factoring,*find the vertex by using AOS formula (x), and subbing x into the original equation for (y).*__In Standard Form:__Once two x-intercepts are found by using the "Quadratic Formula",*find the vertex by using AOS formula (x), and subbing x into the original equation for (y).*

**6. All Forms Connect to Each Other (Vertex, Factored, Standard) **

- In order to find the y-intercept for all of these forms,
*when x is subbed in for 0 (x = 0) the y-intercept is easily obtainable in all forms* - The y intercept is written as
__(0, y)__-*for all forms*

## Assessment

- This assessment
**strengthened my knowledge to understand that their are multiple ways to come to a solution** - In terms of finding the vertex, I
**originally**did it by using**the Quadratic Formula to find the x-intercepts then went along to find the vertex** - I ensured that I was right by
**Completing the Square to the the equation in Vertex Form to obtain the vertex of this quadratic relation** - This made me realize
**there's more than one way of doing a problem involving quadratics and doing it a second time and arriving at the same answer will ensure that your answer is correct**

## Quadratics: Standard Form

**Leaning Quadratics in Standard Form!**

**One Step at a Time!**