16 Quiz Review Quadratic Equations PDF

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All Types of Quadratic Equations Quiz Review...

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REVIEW FOR QUIZ!

NAME: __________________________

Solve each equation (use the method provided) 1. Square Roots Method 2. (x + 3)2 + 2 = –10

3.

Complete the Square. x 2 – 8x + 3 = 0

Factor to solve. x2 – 2x – 15 = 0

4. Quadratic Formula. 6x2 + 2x + 1= 0

Solve each equation. 5. y = 2 (x – 3)2 + 8

7.

y = x2 – 14x + 1

(square roots)

(complete the square)

6.

y = 2x2 + x – 10

8.

y = x2 – 2x + 5

(factor and zero prod)

(quadratic formula)

SOLVING: WHICH METHOD SHOULD YOU USE? Explain why!

1

Equation x + 4x + 3 = 0

A Sq. Roots

B Factor/ZPP

C Complete Sq.

D Quad. Form

2

5x2 – 1 = 6

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

3

x2 – 7x + 1 = 0

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

4

x2 + 10x + 4 = 0

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

5

x2 – 14x = 5

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

6

5 – 3x2 = 20

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

7

x2 + x = 10

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

8

x2 – 4x – 12 = 0

Sq. Roots

Factor/ZPP

Complete Sq.

Quad. Form

2

Solve: Choose which method is best =) !

1.

y = 2 (x + 2)2 + 24

2.

y = x2 – 6x + 8

3.

y = 2x 2 − 5x −12

4.

y = x2 – 12x + 1

! !

Find the discriminant and determine the number and type of solutions. Discriminant 5. y = 3x2 – 3x + 2 6. y = x2 – 10x + 1 7. y = x2 – 4x + 4

Number and Type of Solutions

1.

2.

x2 + 6x + 5 = 0

4(x+2)2 + 100 = 14

3.

4.

x2 – 49 = 0

x2 + 8x + 1 = 10

5.

6.

-2 = 2x2 + 8

x2 + 3x + 1 = 0

7.

8.

4x2 + 12x + 9 = 0

x2 + 6x + 3 = 0

9.

10.

x2 – 36x = 0

x2 + 100 = 0

11.

12.

x2 + 8x + 16 = 3

x2 + 11x = 4

REVIEW PACKET SECTION 1: FACTORING Factor Completely! 1.!!

x2!–!7x!+!6!!

2.!!

x2!–!100!

3.!!

4x2!+!81!

4.!!

25p2!–!16p!

5.!!

m2!–!10m!+!21!

6.!!

y2!–!3y!–!18!!

7.!!!

x2!+!7x!+!12!

8.!!

4x2!+!20x!–!24!

9.!!

4x2!+!20x!+!25!

10.!!

16a2!–!49b2!

11.!!

16x2!+!6x!

12.!!

x2!–!2x!+!1!

13.!

16a2!–!81!!

14.!!

3x2!+!3x!–!36!

15.!

x2!–!8x!+!16!

16.!! !

24z2!–!14z!–!5!

17.!

3m2!–!7m!+!2!

18.!!

3x2!–!75!!

SECTION 2: SOLVING ! Use$the$square$root$method.$ 1.! 5x2!–!7!=!60!

2.!

x2!+!16!=!0!

3.!

4.!

2(x+3)2!+!12!=!4!

! Factor$and$use$the$zero$product$property.! 5.! (2x!+!8)!(x!–!5)!=!0!

6.!

x2!–!2x!+!1!=!0!

7.!

8.!

6x2!+!11x!!=!10!

5x2!+!9!=!134!

x2!+!6x!=!0!

Complete$the$Square.$! 9.! x2!–!4x!–!12!=!0!

10.!

x2!–!2x!–!35!=!0!

12.! ! !

4x2!–!8x!=!40!!

! Use$the$Quadratic$Formula.! 13.! x2!+!5x!–!6!=!0!

14.!

2x2!–!4x!+!3!=!0!

15.!

16.!

10x2!+!9!=!x!

11.!

x2!+!6x!=!23!

2x2!–!x!–!4!=!2!

SECTION 3: GRAPHING Find the vertex of each quadratic function: 1. f(x) = (x+ 2)2 + 5 ( , 3. f(x) = (x – 1)2

(

,

) )

5. f(x) = (x + 10) (x – 2)

4. f(x) = 5x2

(

,

)

(

,

)

(

,

)

(

,

)

6. f(x) = x2 + 2x + 5 (

,

) 8. f(x) = 2x2 + 8x + 5

7. f(x) = 2 (x – 5) (x + 3) ( ! 9.!!!!!!!!y!=!–3 (x – 1)2 + 10 ! ! ! 10.!! y!=!(x + 4)2 + 4! !

2. f(x) = –2x2 – 3

,

)

Opens!Up!!!or!!!Opens!Down!!

!

Stretched,!Shrink,!Standard!

Opens!Up!!!or!!!Opens!Down!!

!

Stretched,!Shrink,!Standard

11. Name 3 synonyms for “solution”: _______________, _______________, _______________ Graph. 2

12.

y = 2 (x + 5 ) − 3

13.

y = − 1 ( x + 5) ( x − 3) 2

14.

y = x 2 + 4x − 6

WRITE THE NEXT STEP ONLY! 1.

(x + 5) 2 = –49

2.

x2 – 9x = 0

3.

2x 2 + 4 = 8

4.

4x2 – 81 = 0

5.

x–2= ± 3

6.

x + 1 = ±6i

7.

Complete the square.

8.

Complete the square.

x 2 – 6x + 10 = 0

9.

Complete the square.

x2 + 8x = 3

10.

x 2 + 10x + 25 = 6

11.

Quadratic Formula. x=

2 ± 9 − 2( −2)( −4 ) 4

Quadratic Formula. x2 + 8x = 3

12.

Quadratic Formula. x=

−10 ± 6i 2 4

WHAT SHOULD YOU DO NEXT? (when solving with square roots or factoring methods) 1.

3.

5.

2x2 + 8 = 10

2.

(x + 4)2 = 25

A.

Divide both sides by 2.

A.

Distribute the square.

B.

Isolate x2.

B.

FOIL.

C.

Square root both sides.

C.

Square root both sides.

x2 – 25x = 0

4.

x2 + 5x + 4 = 0

A.

Factor into (x + 5) (x – 5).

A.

Square root both sides.

B.

Add 25x to both sides.

B.

Subtract 4 from both sides.

C.

Factor out x.

C.

Factor the trinomial.

x2 + 3x = 10

6.

(3x + 1) (x + 4) = 0

A.

Square root both sides.

A.

Set each factor equal to 0.

B.

Subtract 3x from both sides.

B.

FOIL.

C.

Subtract 10 from both sides.

C.

Combine like terms.

! !

7.

2

2x + 7x + 3

WHAT SHOULD YOU DO NEXT in order to factor?$ 8. 9x2 – 30x + 25

A.

List pairs of factors of 3.

A.

Try (3x – 5)2 and check it.

B.

Multiply 2 and 3.

B.

Multiply 9 and 25.

C.

Factor out x.

C.

Set it equal to 0 and solve.

1. x2 + 6x + 5 = 0

2. 4(x+2) 2 + 100 = 14

Factor/ZPP

Sq. Roots Method

3. x 2 – 49 = 0

4. x2 + 8x + 1 = 10

Factor/ZPP

Complete the Square

5. -2 = 2x2 + 8

6. x2 + 3x + 1 = 0

Sq. Roots Method

Quadratic Formula

7. 4x2 + 12x + 9 = 0

8. x2 + 6x + 3 = 0

Factor/ZPP

Complete the Square

9. x2 – 36x = 0

10. x 2 + 100 = 0

Factor/ZPP

Sq. Roots Method

11. x2 + 8x + 16 = 3

12. x 2 + 11x = 4

Complete the Square

Quadratic Formula

Unit 4 QUADRATICS Summary Sheet SECTION 1: FACTORING 1.

Put the polynomial in order of decreasing degree (standard form).

10 + 7x + x2

2.

Factor out the GCF (include any variables!)

4x2 + 14x

All Types

Binomial A2 – B2

If it is a difference of squares, factor into conjugates. Formula: ___________________________________

x2 – 100

Binomial A2 + B2

If it is a sum of squares, the binomial is PRIME.

x2 + 100

If A = 1, 1. List the pairs of factors of C. 2. Find a pair that has a sum/difference of the target #. 3. Write the two binomials.

x2 + 7x + 12

If A = 1, 1. Multiply A and C and list pairs of factors. 2. Find a pair that has a sum/difference of the target #. 3. Factor by grouping. (or factor by trial and error)!

2x2 – 3x – 20

Trinomial x2 + Bx + C

Trinomial

Perfect Square Trinomial

1. 2. 3.

4x2 + 28x + 49

If the first and last terms are perfect squares: Try writing it as a binomial squared. CHECK that the middle term works!!

SECTION 2: GRAPHING A quadratic function is a function with 2 as the highest degree (exponent) Vertex Form 2

y = a (x − h ) + k Vertex: (h, k) 1. a > 0: opens up a < 0: opens down 2. a < -1 or a > 1: stretched -1 < a < 1: compressed 3. Use the squares chart to find other points on the graph.

Intercept Form

Standard Form

y = a(x − p)( x − q )

y = ax 2 + bx + c

⎛ Vertex: ⎜ ⎝

p+q , f 2

( ) ⎟⎠⎞ p+q 2

⎛ −b Vertex: ⎜ , f ⎝ 2a

( ) ⎟⎠⎞ −b 2a

1. Find the x-coordinate of the vertex.

1. Find the x-coordinate of the vertex.

2. Substitute it into the function to find the y-coordinate of the vertex.

2. Substitute it into the function to find the y-coordinate of the vertex.

3. Use the chart to find other points on the graph.

3. Use the chart to find other points on the graph.

Quick Questions. Choose either ANSWER A or ANSWER B. QUESTION ANSWER A What is the form of the function: 1 Intercept Form y = 2x2 + 3x + 2 What is the form of the function: 2 Vertex Form y = 2(x + 3)2 – 10 What is the form of the function: 3 Intercept Form y = – (x + 3) (x – 8) 4 5 6 7 8 9 10 11 12 13 14 15 16

What formula will find the x-coordinate of the vertex for standard form? What formula will find the x-coordinate of the vertex for intercept form? What is the value of C that would complete the square: x 2 – 4x + C

Standard Form Intercept Form Standard Form ⎛b⎞ x =⎜ ⎟ ⎝ 2⎠

−b x= 2a x=

p− q 2

x=

2

p+q 2

4

16

1

2

A binomial sum of squares

A trinomial

Zero Product Property

Complete the Square

Square Roots Method

Quadratic Formula

Square Roots Method

Complete the Square

2

1

0

2

1

0

Real

Imaginary

Real

Imaginary

What is the a-value: y = 2x2 + 5x + 2 What type of polynomial is always prime? What method would you use to solve the equation: y = (x + 3) (2x + 1) What method would you use to solve the equation: y = 4x2 + 10 What method would you use to solve the equation: y = x2 + 10x + 3 The discriminant is 24. How many solutions are there? The discriminant is -10. How many solutions are there? The discriminant is 0. How many solutions are there? The discriminant is -25. What type of solutions are there? The discriminant is 4. What type of solutions are there?

ANSWER B

17

How do you find any x-intercept?

Substitute 0 for x

Substitute 0 for y

18

How do you find any y-intercept?

Substitute 0 for x

Substitute 0 for y

19

What is the quadratic formula?

20

What calculator function can you use to find the vertex of a parabola?

x=

−b b 2 − 4ac 2a

2nd Graph

x=

−b ± b2 − 4ac 2a 2nd TRACE

SECTION 3: SOLVING 1.

Square Roots.

Use When:

2.

An equation has an x2 or (x + c)2 (but does not have an x)

Factor and Zero Product Property.

Use When:

The equation is factorable.

1.

Make sure the equation is in the form: ax2 + bx + c = 0

2

1.

Isolate the x .

2.

Square root both sides.

2.

Factor completely!

3.

Simplify (including the square root!)

3.

Set each factor equal to 0.

4.

Don’t forget the ± sign!

4.

Solve.

5.

Write the solutions together: x = ____, ____

4.

Quadratic Formula.

3.

Complete the Square.

Use When:

1.

The trinomial is not factorable. A=1 and B is even.

Make sure the equation is in the form: Ax2 + Bx = C 2

2.

⎛B ⎞ Use the formula ⎜ ⎟ to determine C. ⎝ 2⎠

3.

Add C to both sides.

4.

Factor the left side of the equation into a binomial squared.

5.

Take the square root of both sides (don’t forget ± )

6.

Isolate the x.

Use When:

The other methods do not apply.

1. Put the equation into standard form: Ax2 + Bx + C = 0 2. Find A, B, C. 3. Substitute A, B, and C into the quadratic formula. Use parentheses! 4. Simplify completely!

Quadratic Formula: x =

−b ± b 2 − 4ac 2a

Discriminant : b2 – 4ac If negative = 2 imaginary solutions If 0 = one real number solution If positive = 2 real number solutions Recall, i =

−1

Quadratic Equations Methods Name: _______________________________________________ I.

What makes an equation a quadratic equation?

II.

There are four methods. List them!

Period: ________

A. B. C. D.

III.

How can you determine which method to use? A.

USE SQUARE ROOTS METHOD IF: If the equation has ______________________ OR ______________________ , (and no ____________)

B.

FACTOR AND USE THE ZERO PRODUCT PROPERTY IF: If the equation has ______________________ AND ______________________ ,

IF THE FIRST TWO METHODS DON’T WORK, CHOOSE BETWEEN THESE TWO: C.

COMPLETE THE SQUARE IF: A=1

D.

AND

the middle term is _________________.

USE THE QUADRATIC FORMULA IF: The middle term is _________________....