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Unit 12: Simplifying Radicals

and The Quadratic Formula

Square Roots and Estimations12.1 Perfect Squares/Cubes12.2 Simplifying Radicals12.3 Simplifying Parts of the Quadratic Formula12.4 The Quadratic Formula12.5 The Quadratic Formula (irrational solutions)12.6 X-intercepts and Vertex

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Square Roots and Estimation

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ESTIMATION CHART

SQUARE ROOT CHART

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Estimate √26

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12. 1 Perfect

Squares/Cubes

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NOTES & PERFECT CUBES

EXAMPLES

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- If the number under the radical is negative, the answer is "no real solution" (*only for square roots*)

- A perfect cube is when the number under the cube root sign has a number multiplied by itself 3 times that equals it.

Ex: ∛8 = ∛2x2x2 so, ∛8 = 2

∛1 = 1

∛8 = 2

∛ = 3

∛ = 4

∛ = 5

∛ = 6

∛ = 7

∛ = 8

∛ = 9

∛ = 10

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C C

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C C

1. √169 2. √-225 3. ∛125

4. ∛-64 5. √100 9 6. √-4 49

7. ∛-8 216 8.√121+ ∛-216 9. 3√25 - ∛1000

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12. 2Simplifying Radicals

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NOTES & Factor Tree

EXAMPLES

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Divisibility Rules

*Multiples of 2: even numbers*Multiples of 3: add the digits & see if they divide by 3*Multiples of 5: ends in a 0 or 5

FACTOR TREE

24 150

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B B

B

AA

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1. √12 2. √35

3. √6

4. 2√90 5. -3√60

B B

B

AA

Simplified:Estimate:

Simplified:Estimate:

Simplified:Estimate:

Simplified:Estimate:

Simplified:Estimate:

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12. 3Simplifying Parts of the Quadratic

Formula

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Vocabulary

EXAMPLES

NOTES

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Quadratic: an equation with one variable that has 2 as the largest exponent

Standard Form of a Quadratic Equation: ax2+bx+c = 0

Parabola: the "U" shaped image you get when you graph a quadratic equation

Vertex:Also known as the point of symmetry of the parabola. (*The minimum or maximum point of the parabola*)

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1. 3x2+2x-4 = 0

a = b = c =

2. 4±√36 2

3. -6±√18 6

4. √32-4(3)(-1) 5. -8±√(8)2-4(1)(7)2(1)

6. -6±√62-4(-2)(-5)2(-2)

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12. 4The Quadratic

Formula

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Notes & Ex 1

EXAMPLES 2-5

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Quadratic Formula:

To solve a quadratic equation using the quadratic formula:

1. Always rearrange the equation into standard form

ax2+bx+c = 02. Identify a, b, and c3. Substitute into the formula using ( )4. Simplify5. Check your answer by plugging in the solution.

1. x2-8x+7 = 0

x = -b ±√b2-4ac 2a

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B/C B/C

B/C B/C

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2. x2-6x+9 = 0 3. 3x2 = 12x-9

4. x2 = -4 5. 2x2-4x = 6

B/C B/C

B/C B/C

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12.5The Quadratic Formula

(irrational solutions)

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NOTES

EXAMPLES

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Steps:

1. Plug in your a, b, & c values.2. This time, the number under the radical will not be a perfect square.3. If possible, simplify the radical.4. Leave your answer in simplest radical form.

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A A

AA

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A A

AA

1. x2+3x+1 = 0 2. x2+4x+2 = 0

3. 7x2-10x-5 = 0 4. x2-6 = -1

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12.6X-intercepts and Vertex

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NOTES

EXAMPLES

NOTES

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Parabola- a graph that makes a "U" shape, like a bowl. The vertex of the parabola is its minimum (low) or maximum (high) point

*We can tell if the parabola will open up or down based on the leading coefficient. If it's positive then the parabola opens UP. If it's negative, then the parabola opens DOWN.

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B B B

A AA

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B B B

A AA

3. y = 5x2 4. y = -3x2+19 5. y = -2x2-8x+12

6. y = 3x2+12x-15 7. y = 4x2+8x+3 8. y = -3x2+6x

Up/Down:Vertex:

Up/Down:Vertex:

Up/Down:Vertex:

Up/Down:Vertex:

Up/Down:Vertex:

Up/Down:Vertex: