algebra 2a - unit 1: factoring and solving quadratics lesson 1-4

which expression is equivalent to (2 - 3i)(5 + i)?

13 - 13i

what are the solutions, if any, of the equation x^2 + 38 = 16?

this equation has no real solutions.

which expression represents the number -3i^4 + 2i^3 + 2i^2 + √-9 rewritten in a + bi form?

-5 + i

the standard form of a complex number is a + bi. what is the standard form for the number 94?

94 + 0i

which expression represents the number -√9 + 14i - i√-4 (4 + 7i) rewritten in a + bi form?

5 + 28i

use the discriminant, b^2 - 4ac, to determine which equation has complex solutions.

x^2 + 3x + 12 = 0

what are the solutions, if any, of the equation (x + 49)^2 = 24?

x = −49 ± 2√6

what are the solutions of the equation x^2 = 25/36?

-5/6, 5/6

what are the zeros of the function f(x) = x^2 − 10x + 21? select all that apply.

3, 7

use the quadratic formula, to solve the quadratic equation 4x^2 + 28x + 49 = 0. which answer provides the correct solution(s) to the equation?

-7/2

what is the factored form of the expression 3x^2 + 6x − 24?

3(x + 4)(x - 2)

which expression or value is equivalent to (-3 + 5i)(-3 - 5i)?

34

what is the factored form of the expression 36x^2 − 49y^2?

(6x + 7y)(6x - 7y)

lesson 2

other methods for solving quadratics

given the complex numbers s and t below, what is s − t? s = 9 + 5i t = 12 - 6i

-3 + 11i

match the numbers that have the same value.

√-49 : 7i √-3 : i√3 √-50 : 5i√2

what is the factored form of the expression x^2 − 7x − 18?

(x + 2)(x - 9)

what are the solutions of the equation (x - 13)^2 = 400?

x = -7 and x = 33

what is the factored form of the expression x^2 − 22x + 121?

(x - 11)^2

examine the complex number 3 + 2i. a: what is the real part of the complex number? b: what is the imaginary part of the complex number?

a: the real part is 3 b: the imaginary part is 2

nando solved the quadratic equation 4x^2 − 24x − 16 = 0 by completing the square as shown. which statements identify nando's mistakes? select all that apply.

in step 7, his final answer should have been x = 3 ± √13. in step 2, he divided 4x^2 - 24x by 4, but he should have also divided 16 by 4. in step 6, he added 3 to the left side of the equation, but he subtracted 3 from the right side

which expression represents the number √-16 - √25 rewritten in a + bi form?

-5 + 4i

the standard form of a complex number is a + bi. what is the standard form for the expression √-64?

0 + 8i

use the quadratic formula, to solve the following equation. 3x^2 - 7x + 4 = 0. what are the solutions to the equation? select all that apply.

4/3, 1

what is the factored form of the expression x^2 − 2x − 15?

(x + 3)(x - 5)

what are the solutions to 3(x + 2)(x − 9) = 0? select all that apply.

-2, 9

solve the quadratic equation −x^2 + 8x − 4 = 0 by completing the square. which expression represents the correct solutions to the equation?

4 ± 2√3

which equations are true? select all that apply.

9x^2 - 1 = (3x - 1)(3x + 1) x^2 - 169 = (x - 13)(x + 13) 16x^2 - 64y^2 = (4x - 8y)(4x + 8y)

what are the solutions, if any, to the equation x^2 - 25 = -7?

x = ±3√2

which expression represents the number 3 - (-7 + 4i) + i(4i^2) rewritten in a + bi form?

10 - 8i

which expression represents the number 12.3 rewritten in a + bi form?

12.3 + 0i

which expression is equivalent to 2i(7 - 4i)?

8 + 14i

which simplifications of the powers of i are correct? select all that apply.

= -1 = -1

all imaginary numbers can be written as complex numbers in the form a + bi. for the number 5i, what is the value of a and b?

a = 0 and b = 5

lesson 1

solving quadratics by factoring

urie completed the square to solve the equation 0 = 3x^2 - 9x + 27 as follows. did urie make a mistake? if so, where?

yes, step 3, he should have added 9/4 to both sides of the equation

what are the zeros of the function f(x) = x^2 − 36?

zeros occur when x = 6 and x = -6

solve the quadratic equation x^2 - 4x - 12 = 0 by completing the square. what are the solutions to the quadratic equation? select all that apply.

x = 6 x = -2

what are the solutions to the quadratic equation 9x^2 + 64 = 0?

x = 8/3i x = -8/3i

use the quadratic formula, to find the solutions to x^2 + 8x + 25 = 0. what are the solutions?

x = -4 ± 3i

what is the factored form of the expression x^2 − 8x − 48?

(x + 4)(x - 12)

what is the factored form of the expression x^2 − 2x − 63?

(x + 7(x - 9)

lesson 3

complex numbers

what are the solutions to the equation (x − 4)(x + 2) = 0? select all that apply.

-2, 4

use the quadratic formula, to solve the equation -2x^2 - 6x + 3 = 0. match each step to the correct method for solving the equation.

step 1: x = 6±√(-6)^2 -4(-2)(3)/2(-2) step 2: x = 6±√36+24/-4 step 3: x = 6±√60/-4 step 4: x = 6 ±2√15/-4 step 5: x = -3±√15/2

what are the solutions to the equation x^2 + 4 = 0?

x = 2i and x = -2i

consider the equation 3x^2 − 2x + 7 = 0 in standard form. which equation shows the coefficients a, b, and c correctly substituted into the quadratic formula?

x = 2±√(-2)^2 - 4(3)(7) / 2(3)

use the quadratic formula, to find the solutions 5x^2 - 6x + 5 = 0

x = 3 ± 4i / 5

use the quadratic formula, to solve the following equation. 5x^2 - 9x - 10 = 0. what are the solutions to the equation? select all that apply.

x = 9 + √281 / 10 x = 9 - √281 / 10

solve the equation x^2 + 21x + 110 = 0 by factoring. what are the solutions to the equation?

-10, -11

examine the complex number 8 - 4i. a: what is the real part of the complex number? b: what is the imaginary part of the complex number?

a: the real part is 8 b: the imaginary part is -4

what are the steps to simplify (4 + 6i)(4 − 6i)? match each step with the correct expression or number.

step 1 : 4(4 - 6i) + 6i(4 - 6i) step 2: 16 - 24i + 24i - 36i^2 step 3: 16 - 36(-1) step 4: 52

what are the steps to simplify (5 + 3i) - (2 + 7i)? match each step with the correct expression.

step 1: 5 + 3i - 2 - 7i step 2: 5 - 2 + 3i - 7i step 3: 3 - 4i

checkpoint: match the numbers that have the same value.

√-36 : 6i √-5 : i√5 √-8 : 2i√2

solve the quadratic equation x^2 − 14x + 24 = 0 by completing the square. what are the solutions to the equation? select all that apply.

2, 12

what is the factored form of the expression 2m^3 − 26m^2 + 80m?

2m(m - 5)(m - 8)

what are the solutions to the equation (x - 32)^2 = 56?

x = 32 ± 2√14

lesson 4

operations with complex numbers

which expression represents the number i^4 + √-81 + i^2 + √-36 rewritten in a + bi form?

0 + 15i

what are the solutions of the equation (x - 9)^2 = 25? select all that apply.

14, 4

ellen solved the equation x^2 − 5 = 59 using the following steps. which statement identifies a mistake ellen made, if any?

in line 4, she forgot to take the negative square root, because squaring both 8 and −8 equals 64.

what is the factored form of the expression 9x^2 + 42x + 49?

(3x + 7)^2

what are the solutions of the equation x^2 + 15 = 79? select all that apply.

8, -8

which expression represents the number -2i(5 - i) + (17 - 8i) rewritten in a + bi form?

15 - 18i

given the complex numbers v and w below, what is v + w? v = 7 + 2i w = 8 - 9i

15 - 7i

solve the quadratic equation −4x^2 + 6x + 16 = 0 by completing the square. which expression represents the correct solutions to the equation?

3 ± √73 / 4

which expression represents the number 9 + √−49 rewritten in a + bi form?

9 + 7i

enrique used the following steps to solve an equation. what mistakes, if any, did enrique make?

in step 3, enrique used the difference of squares pattern incorrectly when factoring.

gemma completed the following steps to find the zeros of the function f(x) = 3x^2 − 6x − 45. what mistakes, if any, did gemma make?

in step 4, when solving for x + 3 = 0, gemma should have subtracted 3 from both sides of the equation to get x = −3.

aurora used the quadratic formula, to solve the equation y = -2x^2 + 6x - 11 as follows. did aurora make a mistake? if so, where?

no, she did not make a mistake.

use the quadratic formula, to find the solutions to 3x^2 - 4x + 7 = 0. what are the solutions?

x = 2 ± i√17 / 3

which expression is equivalent to (7 + 4i) + (8 + i)?

15 + 5i

use the quadratic equation, to solve the equation x^2 + 12x - 7 = 0. match each step to the correct method for solving the equation.

step 1: x = -12±√12^2 -4(1)(-7) / 2(1) step 2: x = -12±√144+28 / 2 step 3: x = -12±√172 / 2 step 4: x = -12±2√43 / 2 step 5: x = -6 ± √43

use the quadratic formula, to solve the quadratic equation 2x^2 + 4x - 1 = 0. what are the solutions to the equation?

x = -2 - √6 / 2 and -2 + √6 / 2

solve the quadratice equation x^2 + 8x - 30 = 0 by completing the square. what are the solutions to the quadratic equation, if any?

x = 4 ± √46

solve the equation 25b^2 − 64 = 0 by factoring. what are the solutions to the equation?

8/5, -8/5

what is the correct order for completing the square to solve x^2 - 2x - 4 = 0? match each step with the correct equation.

step 1: x^2 - 2x = 4 step 2: x^2 - 2x + 1 = 4 + 1 step 3: (x - 1)^2 = 5 step 4: x - 1 = ±√5 step 5: x = 1 ± √5

what are the solutions to the equation -2x^2 - 8x - 22 = 0? use the completing the square method to find the values of x?

x = -2 ± i√7

solve the equation 40 = 2x^2 + 2x by factoring. what are the solutions?

x = -5 and x = 4

use the quadratic formula, to find the solution(s) to 2x^2 - 4x + 27 = 0.

x = 1 ± 5i√2 / 2

use the quadratic formula, to solve the following equation. 9x^2 - 12x + 4 = 0. which option provides the solution(s) to the equation?

x = 2/3

use the discriminant to determine which equation has complex solutions.

x^2 + 2x + 5 = 0

briana used the quadratic formula, to solve the equation 0 = 3x^2 - 2x + 9, as follows. did briana make a mistake? if so, where did the mistake occur, and what was it?

yes, in step 1, she should have had written 2(3) in the denominator.

roman completed the square to solve the equation 0 = -x^2 + 6x - 15 as follows. did roman make a mistake? if so, where?

yes, in step 6, he should have added 3 to both sides of the equation.

consider the equation x^2 + 4x + 9 = 0 in standard form. which equation shows the coefficients a, b, and c correctly substituted into the quadratic formula?

x = -4 ±√(4)^2 -4(1)(9) / 2(1)

what are the solutions to the quadratic equation x^2 + 625 = 0?

x = 25i x = -25i

what are the solutions to the equation x^2 + 4x + 17 = 0.

x = -2 ± i√13

solve the equation −2x^2− 2x + 40 = 0 by factoring. what are the solutions to the equation?

-5, 4

which expression represents the number √-81 - √32 rewritten in a + bi form?

-4√2 + 9i

what are the solutions to the equation x^2 = 49/100? select all that apply.

7/10, -7/10

match each power of i with its equivalent value.

i^8 : 1 i^13 : i i^2 : -1 i^7 : -i

adrian solved the equation x^2 + 9 = 16 using the following steps. which statement identifies a mistake adrian made, if any?

in line 2, he should have subtracted 9 from both sides of the equation because it is not part of the expression that is squared.