Quadratic Formula and Complex Numbers

​REVIEW: Quadratic Functions and Complex Numbers

Algebra 2​

​Equation must be set =0

Identify your a, b, c

Remember the discriminant tells us if the solution will be real or imaginary​

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​Quadratic Formula:

Multiple Choice

For the function below, is the discriminant positive, negative, or zero?

___________

y = x² + 4x + 4

• Positive

• Negative

• Zero

• Not Sure

Multiple Choice

What is the discriminant of -2x² − x − 1 = 0

• 76

• -7

• 9

• none of these

Multiple Choice

If the discriminant is positive, then the solution will be

• one real solution

• two real solutions

• no real solutions

• one imaginary solution

Multiple Choice

A function has a discriminant of -3.
______________
How many x-intercepts does it have?

• 0
• 1
• 2
• 3

Multiple Choice

____________
For the function above, is the discriminant positive, negative, or zero?

• Positive
• Negative
• Zero
• Not Sure

Multiple Choice

What are the x- intercepts?

• x= 0 and x= -4
• x= 0 and x= 4
• y= 0
• x= 2

Multiple Choice

What is the axis of symmetry?

• x= 0 and x=4

• x = -2

• x = 0

• x= 2

Multiple Choice

The coefficient of x^2, the 'a' value of this quadratic, is negative or positive?

• a < 0

• a > 0

Multiple Choice

Does this parabola have a minimum or maximum and what is its value?

• minimum: 2
• maximum: 2
• minimum: 5
• maximum: 5

Vertex: (h, k) min or max

Axis of symmetry: x=h

x-intercepts: roots, solutions, zeros

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Domain: All real numbers

Range: depends on pos. or neg. fxn

neg a--(-infinity, max) or

pos a-- (min, infinity)​

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​Familiarize yourself with vertex form & standard form

​Characteristics of Parabolas

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Multiple Choice

A parabola has a vertex at (-3,2).

What is the axis of symmetry?

• y = -2

• x = 3

• x = -3

• y = 2

Multiple Choice

What is the Range?

• y ≥ 6

• $$y\le6$$  

• -10 ≤ y ≤ 6

• -10 ≤ x ≤ 6

End Behavior: Remember we are looking at what direction the arrows are pointing

left arrow is x-> -∞​

right arrow is x-> ∞​

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​Check your notes

​Interval of Increase/Decrease:

read graph L->R

btwn x-values determine if graph is rising or falling

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Multiple Choice

Describe the end behavior of the graph.

• x → ∞, y→ ∞ and
  x→ ∞, y→⁻∞
• x → ∞, y→ ∞ and
  x→⁻∞, y→∞
• None of these
• x →∞,y→⁻∞ and
  x→∞, y→⁻∞

Multiple Choice

Describe the end behavior of the graph.

• x →∞, y→⁻∞ and x →⁻∞,y→⁻∞
• x →∞, y→∞ and x→⁻∞, y→∞
• x →∞, y→∞ and x→⁻∞, y→0
• x →∞,y→∞ and x→⁻∞, y→⁻∞

Multiple Choice

What are the values of the constants a, b, and c in the given function?

$$f\left(x\right)=2x^2-3x+4$$  

• a = 2; b = 3; c = 4

• a = 2; b = -3; c = 4

• a = 2; b = -3; c = -4

• cannot be determined

Multiple Choice

Which of the following is the correct formula in determining the vertex of the quadratic function?

• $$x=-\frac{b}{2a}$$  

• $$x=-\frac{c}{2a}$$  

• $$x=\frac{b}{2a}$$  

• $$x=-\frac{b}{2c}$$  

Fill in the Blank

$$f\left(x\right)=-2x^2+8x-7$$  

What is the opening of the given function?

Multiple Choice

$$f\left(x\right)=a\left(x-h\right)^2+k$$  

What is the vertex of the given quadratic function?

• V(0,k)

• V(h,0)

• V(h,k)

• V(k,h)

Multiple Choice

Which equation best represents the graph?

• y = 2(x - 4)² + 5
• y = 2(x + 4)² + 5
• y = -2(x - 4)² + 5
• y = -2(x + 4)² + 5

Multiple Choice

Identify the domain of the function.

• $$\left(-\infty,\ \infty\right)$$  

• $$\left[2,\ \infty\right)$$  

• $$\left[-3,\ \infty\right)$$  

• $$\left(-\infty,\ -3\right]$$  

Multiple Choice

Identify the range of the function.

• $$\left(-\infty,\ \infty\right)$$  

• $$\left[2,\ \infty\right)$$  

• $$\left[-3,\ \infty\right)$$  

• $$\left(-\infty,\ -3\right]$$  

Multiple Choice

Identify the interval of decrease.

• $$\left(-\infty,\ 2\right)$$  

• $$\left(2,\ \infty\right)$$  

• $$\left[-3,\ \infty\right)$$  

• $$\left(-\infty,\ -3\right]$$  

Remember Like Terms:

real with real

imaginary with imaginary

i²​=-1

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​Review rules with your notes

​Complex Numbers

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Multiple Choice

What determines if a quadratic will have complex solutions?

• The a-value

• The discriminant is a perfect square

• The discriminant is negative

• The x-intercepts are decimals

Multiple Choice

Simplify  $$\sqrt[]{-25}$$  

• -5i
• 5
• 5i
• -5

Multiple Choice

Simplify  $$\sqrt[]{-32}$$   

• $$4i\sqrt[]{2}$$  

• $$2i\sqrt[]{4}$$  

• $$4\sqrt[]{2i}$$  

• $$2\sqrt[]{4i}$$  

Multiple Choice

If a quadratic has a vertex at the point (1,1), and opens up, how many x-intercepts does it have?

• 0
• 1
• 2
• 3

Multiple Choice

(4 - 2i) - (3 + 6i)

• 7 - 4i

• 1 + 4i

• 1 - 8i

• 7 - 8i

Multiple Choice

(2i)(3i)

• 5i
• -5
• 6i
• -6

Multiple Choice

Simplify $$\frac{4-9i}{4+14i}$$  

• $$\frac{-7}{53}+\frac{49i}{106}$$  

• $$\frac{4}{15}-\frac{3i}{5}$$  

• $$-\frac{55}{106}-\frac{23i}{53}$$  

• $$\frac{4}{21}-\frac{3i}{7}$$  

Multiple Choice

Simplify   $$\frac{8+5i}{1-7i}$$  

•   $$\frac{-3+76i}{65}$$  

•   $$\frac{11+77i}{50}$$  

•   $$\frac{-11-5i}{7}$$  

• $$\frac{-27+61i}{50}$$  

Always check for GCF, factor it out first

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Remember ac method:

Find the factors of "a*c" that sum to "b"​

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​Factoring Quadratics

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Multiple Choice

Factor and solve on paper. Then choose the correct solutions.

x² - 7x + 10 = 0

• x = -2 and x = -5

• x = 2 and x = 5

• x = 1 and x = -7

• x = -1 and x = 10

Multiple Choice

Factor:
x² + 5x - 24

• (x - 8)(x + 3)
• (x + 8)(x - 3)
• (x + 6)(x - 4)
• (x + 12)(x - 2)

Multiple Choice

3v² - 8v + 5

• (3v + 5)(v - 1)

• (3v - 5)(v + 1)

• (3v - 5)(v - 1)

• (3v + 5)(v + 1)

Multiple Choice

Find the factors

16xy - 24y

• 4y (4x - 6)

• 8y (2xy - 3y)

• 8y (2x - 3)

• 2xy (8 - 12y)

Multiple Choice

Factor out the GCF

3x³ + 9x

• 3x (x² + 3)

• 3x ( x³ + 9)

• x (3x² + 9)

• 3x (x³ + 3x)

Multiple Select

Which is an example of difference of two squares?

• ( x³ - 4)

• ( x² - 3)

• ( x² + 4)

• ( x² - 4)

Multiple Choice

Factor 4x²-9

• (4x-9)(4x+9)

• (2x-3)(2x+3)

• 2x-3

• (2x-4.5)(2x+4.5)

​Always set your standard form equation = 0

​ax² + bx + c = 0

Once you have the factors, set EACH factor =0, and solve for variable

These are the solutions (zeros) to the quadratic​

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​Zero Product Property

Multiple Choice

Solve using the zero product property

(x+2)(x-6)=0

• x=2, -6

• x=-2, -6

• x=-2, 6

• x=2, 6

Multiple Choice

Solve using the zero product property

(2x-7)(x+2)=0

• x=7, -2

• x=-7/2, 2

• x=7, 2

• x=7/2, -2

Multiple Choice

Solve using the zero product property

x(2x-3)=0

• x=0, 3/2

• x=0, -3/2

• x=0, -2/3

• x=0, -2/3

Multiple Choice

What do the "Zeros" represent on a quadratic equation graph?

• x-intercepts

• y-intercepts

• vertex

• focus

Multiple Choice

Solve using the Zero-Product Property.

7x(x − 6) = 0

• x =7 and x = −6

• x = 0 and x = 6

• x = 0 and x = −6

• x = 7 and x = 6

Multiple Choice

Factor and Solve: Remember the two numbers you choose when factoring, are not your solutions to the equation!

a² + 2a − 24 = 0

• a = −6, 4

• a = 6, −4

• a = 12, −2

• a = 8, −3

Multiple Choice

x² + 4x - 32 = 0

• x= -8, 4

• x= 8, -4

• x= -8, -4

• x = -2, 16

Multiple Choice

Solve by Factoring Completely:  3x² + 18x +15 = 0 Hint: Factor GCF first.

• x=5 x=-1

• x=3 x = 1

• x= -5 x= - 1

• x= -5 x = 1

Multiple Choice

Solve by factoring: x² = 9x - 20

• x = -4 and x = -5

• x = 20 and x = -9

• x = 4 and x = 5

• x = -4 and x = 5

Multiple Choice

Solve using the zero product property

(x-5)(x-10)=0

• x=5, 10

• x=-5, -10

• x=-5, 10

• x=5, -10

Remember this method only works if a=1

if your x²​ coefficient is >1, you need to divide every term by the "a" value

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​Review Perfect Square Trinomials

​Solve Quadratics by Complete the Square Method

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Multiple Choice

What is the first step to solving THIS equation by completing the square?
a² + 10a + 21 = 0

• Set the equation equal to zero
• Divide 10 by 2 and add the result to both sides
• Add a² and 10a together
• Subtract the 21

Multiple Choice

Solve $$\left(x+3\right)^2-13=0$$  , leave your answer in the form of $$p\pm\sqrt[]{q}$$  

• $$x=-3\pm\sqrt[]{13}$$  

• $$x=3\pm\sqrt[]{13}$$  

• $$x=-3\pm\sqrt[]{-13}$$  

• $$x=-3-\sqrt[]{13}$$  

Multiple Choice

Complete the Square
x² + 6x = 5

• (x + 3)² = 5
• (x + 6)² = 9
• (x + 3)² = 14
• (x + 6)² = 14

Multiple Choice

Solve by completing the square:
k² − 12k + 23 = 0

• {6 + √13, 6 - √13}
• {-6 + √13, -6 - √13}
• {6 + √59, 6 - √59}
• {-6 + √59, -6 - √59}

Multiple Choice

Solve by completing the square.
y² + 10y = -9

• 1 and -12
• -1 and -9
• 1 and -9
• 1 and -1

Multiple Choice

Solve using the zero product property

(x+13)(x+7)=0

• x=13, 7

• x=-13, -7

• x=-13, 7

• x=13, -7

Multiple Choice

Solve using the zero product property

(x-9)(x+1)=0

• x=9, 1

• x=-9, 1

• x=-9, -1

• x=9, -1

Multiple Choice

Solve using the zero product property

(x+5)(x-11)=0

• x=5, -11

• x=-5, -11

• x=-5, 11

• x=5, 11

Multiple Choice

Solve using the zero product property

(2x-7)(x+2)=0

• x=7, -2

• x=-7/2, 2

• x=7, 2

• x=7/2, -2

Multiple Choice

Solve using the zero product property

(3x+1)(3x-1)=0

• x=3, -3

• x=-1/3, 1/3

• x=3

• x=3/1, -3/1

Multiple Choice

Solve using the zero product property

(3x+4)(2x+5)=0

• x=-4/3, -5/2

• x=-3/4, -2/5

• x=3/4, 2/5

• x=4/3, 5/2

Multiple Choice

Solve using the zero product property

(x-2)(2x+3)=0

• x=-2, 3/2

• x=2, -2/3

• x=2, -3/2

• x=-2, 2/3

Multiple Choice

Solve using the zero product property

(3x-1)(x+7)=0

• x=1/3, -7

• x=3/1, -7

• x=-1/3, 7

• x=-3/1, 7

Multiple Choice

Solve using the zero product property

x(2x-3)=0

• x=0, 3/2

• x=0, -3/2

• x=0, -2/3

• x=0, -2/3

Multiple Choice

Solve using the zero product property

5x(x+15)=0

• x=0, 15

• x=0, -15

• x=-5, -15

• x=5, 15