# Unit # 3 - Standard Form

### by Joy Mateo

## Introduction

**y=ax²+bx+c**

- the value of
**a**tells you the**shape**and**direction**of the opening - the value of
**c**is the**y-intercept** **solving**using the**quadratic formula**produces the**x-intercepts****completing the square**gets**vertex form**, which tells you if it is a**max or min**

## Quadratic Formula

*but*using the

**Quadratic Formula always works!**

The Quadratic Formula is derived from completing the square, where the a, b, and c are all numbers, and a ≠ 0

To start we have ax²+bx+c=0 [ The equation must be **set to 0**!]

Which after many steps of isolation and simplifying, we end up with the** quadratic formula:****x= -b±√b****²-4ac-------------------**

**2a**

Now that we have the formula, all you need to do is plug in the values and solve!

Start by differentiating your a, b, and c values so you don't confuse them!__ Remember to keep the sign with it__

Remember the sign **± **means that __we will add one time and subtract anothe__r so we end up with two possibly different solutions (for the two x intercepts)

Also __pay attention to integers and BEDMAS!__

The answer will result in an **Exact Solution/Root** expressed with a square root sign (unless the number under it is a perfect square then it is factorable!)

For an **Approximate solution**, put the numbers into your calculator and round as indicated.

- The x-coordinate of the vertex of a parabola is -b/2a
- The equation for the axis of symmetry is x= -b/2a

**Ex.**

2x² + 9x + 6 = 0

a= 2

b= 9

c= 6

1) x= -b±√b²-4ac

-------------------

2a

2) x= -9±√81-4(2)(6)

-------------------

2(2)

2) x= -9±√81-48

-------------------

4

3) x= -9±√33

-------------------

4

4) x= -9+ √33 or x= -9-√33 [EXACT]

------------------- ------------------

4 4

5) x = -0.81 or x=-3.69 [APPROXIMATE]

## Completing the Square

**Steps:**

1) Bracket the first two terms and factor them

2) Take half of the second term and square it

3) Add and subtract it into the equation

4) Factor the perfect square trinomial

5) Multiply the GCF with the number outside the bracket

6) Subtract/add the terms outside

- 7) Vertex = opp sign inside the bracket (x) and the outside number (y)
- the y value tells you if it is a max/min value meaning it opens down/up

**Ex. **

1. y= -3x² + 12x - 13

y= (-3x² + 12x) - 13

------------------------

-3

y= -3(x²-4x) -13

2. 4/2² = 2² = 4

3. y= -3(x²-4x+4) -4-13

4.y= -3(x-2)² -4-13

5.y= (x-2)² 12-13

6.y= (x-2)² -1

7) Vertex = (+2, -1)

## Discriminant

__under the square root__in the quadratic formula. It tells you

__how many real solutions a quadratic equation has__

Here is the__ formula__ for the__ discriminan__t:**D = b****²-4ac**

- D>0 = two real solutions
- D<0 no real solution
- D= 0 one real solution

From an equation given in standard form, we can differentiate the a, b, and c values and plug them into our discriminant equation to determine how many solutions there will be.**Ex.**

9x² -12x+4= 0

a =9

b= -12

c=4

d=b² - 4ac

d= (-12)² - 4(9)(4)

d=144 - 144

d= 0

- One real solution!

## Word Problem

## Reflection

- Vertex form is most preferable for finding the vertex and determining if it is a min/max value
- Factored form is most preferable for determining the roots (x-intercepts)
- Standard form is most preferable for its versatility, great for combining other equations because a,b,c are all real numbers