Algebra II Unit 5 Answers PHS

(L6) Add the complex numbers. (9+7i)+(-5-3i)=

4+4i

(L7) Multiply the complex numbers. (2-i)(2+i)

5

(Q2) Add the complex numbers. (4-2i)+(2+6i)

6+4i

(L7) The conjugate of 7-2i is ___.

7+2i

(PT) Add the complex numbers. (5-2i)+(2+4i)

7+2i

(L1) Combine the equations into one standard form equation. 7x+3=5 and y-1=6

7x-y=-5

(L6) When we combine complex numbers, we combine the ___ parts, then combine the imaginary parts.

real

(Q2) When we combine complex numbers, we combine the ___ parts, then combine the imaginary parts.

real

(PT) If an inequality contains the symbols, ≤ or ≥ it would be graphed as a ___ line.

solid

(PT) Any value or values for a variable or variables that make an equation or inequality true is called a ___.

solution

(Q2) Find the real roots for y=x²+6x-7. Use the quadratic formula x=-b±√b²-4ac/2a.

(1,0) (-7,0)

(L4) Find the real roots for y=2x²-8x+6. Use the quadratic formula x=-b±√b²-4ac/2a.

(1,0) (3,0)

(PT) Factor the quadratic inequality to find the x-intercepts. y≤x²+2x-8

(2,0) (-4,0)

(L8) Find the solution(s) to the systems of equations algebraically. (Use quadratic formula) {y=-x+3 {y=2x²-3x+4

(2,4)

(PT) Find the vertex for y=x²-6x+5. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(3,-4)

(PT) Find the solution(s) to the systems of equations algebraically. (Use substitution and factoring) {y=x2-4x+2 {y=x-2

(4,2) (1,-1)

(PT) Find the real roots for x²-6x+5. Use the quadratic formula x=-b±b²-4ac/2a. Select TWO that apply.

(5,0) (1,0)

(PT) Using the inequality y≤x²+2x-8 and the x-intercepts(-4,0),(2,0) to answer the following:

(A) distance between the x-intercepts: 6 (B) x-coordinate of the vertex: 1 (C) y-coordinate of the vertex: -9

(Q2) Complete the table. y=x²+6x-7

-12 -12 -7 -7

(L7) Multiply the complex numbers. (3+4i)(-2+2i)

-14-2i

(PT) Multiply the complex numbers. (5+i)(-3+2i)

-17+7i

(Q3) Multiply the complex numbers. (-7+i)(3-2i)

-19+17i

(L9) The possible number of intersection points of two different ellipses range from ___ to as many as four.

0

(L7) Mr. Silverstone invested some money in 3 different investment products. The investment was as follows: A. The interest rate of the annuity was 4%. B. The interest rate of the annuity was 6%. C. The interest rate of the bond was 5%. D. The interest earned from all three investments together was $950. Which linear equation shows interest earned from each investment if the total was $950?

0.04a+0.06b+0.05c=950

(Q3) Multiply the complex numbers. (1+i)(2+i)

1+3i

(Q3) Multiply the complex numbers. (2-4i)(2+4i)

20

(L7) Divide the complex numbers. 4+6i/3+2i

24/13 + 10/13i

(L7) Divide the complex numbers. 6+3i/7-5i

27/74 + 51/74i

(L7) Find the radius of the circle. x²+2x+y²+8y+8=0

3

(L4) Find the values of a, b, and c for y<x²+1.

a=1 b=0 c=1

(L5) Solve. a²+2=0

a=±i√2

(PT) Find the answer as an imaginary number. a²=-2

a=±i√2

(L9) When finding the solution to a system of equations, it is important to find ___ solutions.

all

(Q3) When finding the solution to a system of equations, it is important to find ___ solutions.

all

(Q3) The conjugate is used in ___ of complex numbers.

division

(Q3) The ___ formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant, b2-4ac, is positive or zero.

quadratic

(L3) Find the equations of the asymptotes of the hyperbola. (x-2)²/4² - (y-1)²/2² =1 Use the formula y=±b/a(x-x₁)+y₁.

y=0.5x y=-0.5x+2

(Q2) Find the answer as an imaginary number. y²=-6

y=±√6i

(PT) Find answers as an imaginary number. z²+7=0

z=±i√7

(L8) If the discriminant is ___, there will be one real number root and the vertex of the quadratic will be on the x-axis.

zero

(PT) Divide the complex numbers. 3+9i/8+i

³³/₆₅ + ⁶⁹/₆₅i

(Q3) Divide the complex numbers. 10+i/8-2i

³⁹/₃₄ + ⁷/₁₇i

(PT) Find the values of a, b, and c for y=x²-6x+5.

a= 1 b= -6 c= 5

(Q2) Find the values of a, b, and c for y=x²+6x-7.

a= 1 b= 6 c= -7

(PT) (n) ___ number is a number of the form a+bi where a and b are real numbers and and i²=-1.

complex

(L7) The ___ of a complex number a+bi is the complex number a-bi

conjugate

(PT) The ___ of a complex number a+bi is the complex number a-bi.

conjugate

(Q3) The ___ of a complex number a+bi is the complex number a-bi.

conjugate

(L1) If an inequality contains the less than symbol (<), its graph would be a ___ line.

dashed

(L1) If you drew the inequality y<(x-3)²+2 on a graph, would the line be dashed or solid.

dashed

(PT) If an inequality contains the less than symbol (<) or greater than symbol (>), its graph would be a ___ line.

dashed

(Q1) If an inequality contains the less than symbol (<) or greater than symbol (>), its graph would be a ___ line.

dashed

(Q1) If the coefficient in front of the x² in a quadratic equation is negative, the parabola will curve ___.

down

(L9) To find the value of y using the value of x, use ___ equation of the system.

either

(Q3) To find the value of y using the value of x, use ___ equation of the system.

either

(L8) The quadratic formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant, b²-4ac, is ___.

either positive or zero

(L5) Given: f(x)=5x;g(x)=x-3 f(g(x))=?

f(g(x))=5x-15

(L6) Find the inverse of the function. f(x)=3x+2

f¯¹(x)=⅓x-⅔

(Q2) A ___ is a quantity that has both length and direction.

vector

(L9) Find the solution to the system of equations. {2x-4y-3z=2 {x-2y+z=-9 {-3x+y+z=-4

x= 1 y= 3 z= -4

(Q1) Factor the quadratic inequality to find the x-intercept points. y≥x²-2x-3

x=-1 x=3

(L2) Factor the quadratic inequality to find the x-intercepts. y≥x²+2x-8

x=2 x=-4

(L2) Factor the quadratic equation to find the x-intercept points. y=x²-2x-8 _____ and _____

x=4 x=-2

(Q2) Find the answer as an imaginary number. x²+9=0

x=±3i

(L5) Solve. x²=-25

x=±5i

(L5) Solve. 3x²-2x+5=0

x=⅓±√¹⁴/₃ i

(PT) Complete the table. y=x²-6x+5

y=-3 y=-3 y=5 y=5

(L4) Complete the table. y=-2x²-10x-12

y=-4 y=-12 y=-12 y=-4

(PT) y=x²+2x-8

y=-8 y=-8 y=-5 y=-5

(PT) A(n) ___ number is a number of the form bi where b is a real number, and i²=-1.

imaginary

(Q2) A(n) ___ number is a number of the form bi where b is a real number, and i²=-1.

imaginary

(Q2) Complex numbers are represented on a Cartesian coordinate system with a horizontal real axis and a vertical ___ axis.

imaginary

(L9) Find the product of the matrices. A=[2 3] .....[4 7] and B=[⁷/₂ -³/₂] .....[-2 1]

[1 0] [0 1]

(L4) Multiply. ..[1 0 5 1] 2[0 4 9 -6] ..[0 -6 6 -3]

[2 0 10 2] [0 8 18 -12] [0 -12 12 -6]

(L3) Add the matrices. [5 2 1 -4]+[-2 -2 -5 2] [1 -1 3 0]+[-4 0 -7 0] [-3 6 -2 3]+[3 -6 4 -3]

[3 0 -4 -2] [-3 -1 -4 0] [0 0 2 0]

(L7) Multiply the complex numbers. (-5-3i)(4-2i)

-26-2i

(Q3) Find the solution(s) to the systems of equations algebraically. {y=x2+4x-1y=x-3 (Use factoring.)

(-1,-4) (-2,-5)

(L9) Find the solution(s) to the system of equations. (Use substitution and factoring. Use the linear equation to find y -values) {x²+y²=5 {y=x+1

(-2,-1) (1,2)

(L8) Find the solution(s) to the systems of equations algebraically. (Use factoring) {y=4x-2 {y=3x²+7x-8

(-2,-10) (1,2)

(L4) Find the real roots for y=-2x²-10x-12. Use the quadratic formula x=-b±√b²-4ac/2a.

(-2,0) (-3,0)

(L4) Find the vertex for y=-2x²-10x-12. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(-2½,½,)

(Q2) Find the vertex for y=x²+6x-7. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(-3,-16)

(L3) Find the real roots for y=x²+6x+9. Use the quadratic formula x=-b±√b²-4ac/2a.

(-3,0)

(L3) Find the vertex for y=x²+6x+9. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(-3,0)

(Q1) Use the quadratic formula, x=-b±√b²-4ac/2a to find the real roots of the quadratic equation. x²-3x-18

(-3,0) (6,0)

(L4) Find the real roots for y=x²+3x-4. Use the quadratic formula x=-b±√b²-4ac/2a.

(-4,0) (1,0)

(L6) Find the solution to the system of equations. {4x+5y=-6 {3x+2y=-8

(-4,2)

(L9) Find the solution(s) to the system of equations. Look at the graph for the number of solutions. Select all that apply. (Use substitution) {y=x-2 {y=x²-4x-2

(0,-2) (5,3)

(L9) Find the solutions to the system of equations. Look at the graph for the number of solutions. Select all that apply. (Use substitution. Use the parabola equation to find the value for y. ) {y=x² {x2+(y+2)²=4

(0,0)

(L8) Find the solution to the linear system. {x+2y=-3 2x-3y=8

(1,-2)

(Q3) Find the solution(s) to the systems of equations algebraically. {y=x2+x-2y=-x+1 (Use substitution and quadratic equation) x=-b±b2-4ac2a

(1,0) (-3,4)

(L8) Find the solution to the linear system. {x-2y=4 {4x+2y=6

(2,-1)

(L6) Subtract the complex numbers. (-5+3i)-(-1+2i)

-4+i

(Q3) Divide the complex numbers. 2+9i/3-7i

-⁵⁷/₅₈+⁴¹/₅₈i

(L7) Find the determinant of the matrix. [12] [36]

0

(L2) Choose True or False False True False

Choose True or False If the coefficient in front of the x² in a quadratic equation is negative, the parabola will curve up. If the parabola intersects the x-axis at two points, the axis of symmetry will be halfway between the two points of intersection. Factoring should always be used to solve quadratic equations.

(PT) Find the complex number. Real: -3 Imaginary: -5i

Complex Number: -3-5i

(PT) Find the complex number. Real: 12 Imaginary: 2i

Complex Number: 12+2i

(Q2) Find the complex number. Real: -11 Imaginary: -5i

Complex: -11-5i

(L5) Find the complex number. Real: 11 Imaginary: -3

Complex: 11-3i

(L5) Find the complex number. Real: 2 Imaginary: i

Complex: 2+i

(L5) Find the complex number. Real: 5 Imaginary: -6√-1

Complex: 5-6i

(Q2) Find the complex number. Real: 7 Imaginary: 9i

Complex: 7+9i

(Q1) ______ is not a technique that should be used every time to solve quadratic equations.

Factoring

(L5) Identify each as a Real or Complex number. 2+3i² -9-2i 7+4i

Identify each as a Real or Complex number. Real number Complex number Complex number

(L8) Find the solution(s) to the systems of equations algebraically. (Use quadratic equation.) {y=-x+3 {y=2x²-3x+4

No real solution

(L4) Find the slope and y -intercept. -x+3y>6

Slope (m):⅓ y-Intercept (b):2

(Q2) The quadratic ___ calculates the roots of a quadratic equation and indicates the nature of its graph.

formula

(L1) Any point that is on a ___ is a solution.

graph

(L8) If the discriminant is ___, a quadratic will have no real number roots and will not intersect the x-axis at all.

negative

(PT) If the discriminant is ___, a quadratic will have no real number roots and will not intersect the x -axis at all.

negative

(L8) The graphs of a line and parabola could intersect at one point, two points, or ___.

no points

(Q3) The graphs of a line and parabola could intersect at one point, two points, or ___.

no points

(L8) If the discriminant, b²-4ac, is ___, a quadratic will have two real roots, two points of intersection with the x-axis.

positive

(PT) A ___ is a quantity that has both length and direction.

vector

(L5) Multiply the matrices. ................[1 3 2] [1, 0, 3]•[0 -1 4] ................[-2 0 1]

[-5, 3, 5]

(L3) Find the values of a, b, and c for y=x²+6x+9.

a= (E.) 1 b= (A.) 6 c= (C.) 9

(L3) Find the values of a, b, and c for y=x²-6x.

a= 1 b= -6 c= 0

(L3) Find the values of a, b, and c for y=x²-9x+18.

a= 1 b= -9 c= 18

(L3) Find the values of a, b, and c for y=x²+5x+7.

a= 1 b= 5 c= 7

(L4) Find the values of a, b, and c for y<-2x²-10x-12.

a=-2 b=-10 c=-12

(L4) Find the values of a, b, and c for y=x²+3x-4.

a=1 b=3 c=-4

(L4) Find the values of a, b, and c for y=2x²-8x+6.

a=2 b=-8 c=6

(L3) The quadratic formula can be used to find the roots of ___ quadratic equation.

any

(Q1) The quadratic formula can be used to find the roots of ___ quadratic equation.

any

(L5) A(n) ___ number is a number of the form a+bi where a and b are real numbers and i²=-1.

complex

(Q2) A(n) ___ number is a number of the form a+bi where a and b are real numbers and i²=-1.

complex

(L4) The quadratic ___ calculates the roots of a quadratic equation and indicates the nature of its graph.

formula

(Q1) Any point that is on a ___ is a solution.

graph

(Q1) If the parabola intersects the x -axis at two points, the axis of symmetry will be ___ between the two points of intersection.

halfway

(L5) A(n) ___ number is a number of the form bi where b is a real number, and i²=-1.

imaginary

(L6) Complex numbers are represented on a Cartesian coordinate system with a horizontal real axis and a vertical ___ axis.

imaginary

(L4) If the value under the square root sign in the quadratic equation is negative, there are ___ x-intercepts.

no

(L1) Is (2,6) a solution to its equation? Is (4,2) a solution to its equation?

no no

(L4) Find the real roots for y<x²+1. Use the quadratic formula x=-b±√b²-4ac/2a.

no real number roots

(L3) Find the real roots for y=x²+5x+7. Use the quadratic formula x=-b±√b²-4ac/2a.

no real roots

(L4) To find the solution area of the graph of an inequality, chose a point ___ the curve and determine if it is a part of the solution.

not on

(L3) The ____ formula, x=-b±√b²-4ac/2a is used to find the solutions to a quadratic equation of the form ax²+bx+c=0.

quadratic

(L3) The solutions to a quadratic equation are called the ___ or x-intercepts.

roots

(Q1) The solutions to a quadratic equation are called the ___ or x-intercepts.

roots

(L2) Find the slope and y-intercept. 4x+3y=12

slope: -⁴/₃ y-intercept: 4

(L1) If an inequality contains the symbols, ≤ or ≥ it would be graphed as a ___ line.

solid

(Q1) If an inequality contains the symbols, ≤ or ≥ it would be graphed as a ___ line.

solid

(L1) Any value or values for a variable or variables that make an equation or inequality true is called a ___.

solution

(Q1) Any value or values for a variable or variables that make an equation or inequality true is called a ___.

solution

(L6) A ___ is a quantity that has both length and direction.

vector

(L4) Complete the table. y=x²+1

y=2 y=2 y=5 y=5

(L4) Complete the table. y=2x²-8x+6

y=6 y=6

(L1) Is (-2,9) a solution to its equation? Is (1,8) a solution to its equation?

yes no

(L2) Find the coordinates of the vertex of the parabola. y²+4y-2x=6

(-5,-2)

(L4) Find the vertex for y=x²+3x-4. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(-³/₂,²⁵/₄)

(L3) Find the vertex for y=x²+5x+7. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(-⁵/₂,³/₄)

(L5) Find the center of the ellipse. x²/49 + y²/25 =1

(0,0)

(L4) Find the vertex for y=x²+1. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(0,1)

(Q1) Use the quadratic formula, x=-b±√b²-4ac/2a to find the real roots of the quadratic equation. x²+4x-5

(1,0) (-5,0)

(L4) Find the vertex for y=2x²-8x+6. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(2,-2)

(L1) Find the vertex equation by completing the square. y=3x²-12x+6

(2,-6)

(Q1) Find the vertex by completing the square. y=x²-4x-5

(2,-9)

(L3) Find the real roots for y=x²-9x+18. Use the quadratic formula, x=-b±√b²-4ac/2a.

(3,0) (6,0)

(L1) Find the coordinates of the vertex of the parabola. y=(x-5)²+2

(5,2)

(L3) Find the real roots for y=x²-6x. Use the quadratic formula x=-b±√b²-4ac/2a.

(6,0) (0,0)

(Q1) Using the inequality y≥x²-2x-3 and the x-intercepts (-1,0), (3,0) answer the following questions:

(A) distance between the x-intercepts: (D.) 4 (B) x-coordinate of the vertex: (E.) 1 (C) y-coordinate of the vertex: (C.) -4

(L2) Using the equation y≥x²+2x-8 and the x-intercepts (-4,0),(2,0) answer the following:

(A) distance between the x-intercepts: 6 (B) x-coordinate of the vertex: -1 (C) y-coordinate of the vertex: -9

(L2) Using the equation y=x²-2x-8 and the x-intercepts (-2,0),(4,0) answer the following:

(A) distance between the x-intercepts: 6 (B) x-coordinate of the vertex: 1 (C) y-coordinate of the vertex: -9

(L4) Transform the equation into the standard form for a hyperbola: 4x²-9y²+8x+18y-41=0

(x+1)²/9 - (y-1)²/4 =1

(L6) Which is the standard form equation of the ellipse? 8x²+5y²-32x-20y=28

(x-2)²/10 + (y-2)²/16 =1

(L3) Find the vertex for y=x²-9x+18. Use (-b/2a , -(b²-4ac)/4a) from the quadratic formula x=-b±√b²-4ac/2a.

(⁹/₂,-⁹/₄)

(L6) Add the complex numbers. (-6-i)+(-4-5i)=

-10-6i

(L3) |x+3|<7

-10<x<4

(Q1) Complete the table. y=x²-2x-3

-3 -3 5 5

(L1) Using the equation, complete the table. y=3x²-12x+6 vertex (2,-6)

-3 -3 6 6

(L6) Subtract the complex numbers. (3-2i)-(7+6i)

-4-8i

(L2) Complete the table. y=x²-2x-8 (x=-3, x=1)

-5 -5

(L2) Complete the table. y=x²-2x-8 (x=3, x=-1)

-5 -5

(Q1) Using the equation, complete the table. y=x²-4x-5 vertex (2,-9)

-5 -5 7 7

(L2) Find the transpose of the matrix. .....[1 3 5 7] Y=[0 1 8 9] .....[-1 5 9 0]

.....[1 0 -1] Yᵀ[3 1 5] .....[5 8 9] .....[7 9 0]

(L6) Add the complex numbers. (4+2i)+(-2+3i)=

2+5i