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Questions and Answers
What is the factored form of the expression $x^2 + 13x + 42$?
What is the factored form of the expression $x^2 + 13x + 42$?
- (x + 5)(x + 8)
- (x + 3)(x + 10)
- (x + 6)(x + 7) (correct)
- (x - 6)(x - 7)
What is the factored form of the expression $9x^2 - 24x + 16$?
What is the factored form of the expression $9x^2 - 24x + 16$?
- (3x - 4)(3x - 4)
- (3x + 4)(3x + 4)
- (9x - 16)(x + 1)
- (3x - 4)^2 (correct)
What are the solutions to the equation $x^2 + 3x - 28 = 0$?
What are the solutions to the equation $x^2 + 3x - 28 = 0$?
4, -7
What are the zeros of the function $f(x) = x^2 + 8x + 16$?
What are the zeros of the function $f(x) = x^2 + 8x + 16$?
What are the solutions of the equation $(x + 16)^2 = 28$?
What are the solutions of the equation $(x + 16)^2 = 28$?
Did Anna make a mistake when solving the equation $(x + 4)^2 = 2$?
Did Anna make a mistake when solving the equation $(x + 4)^2 = 2$?
Solve the quadratic equation $x^2 + 3x - 10 = 0$ by completing the square. What is the solution?
Solve the quadratic equation $x^2 + 3x - 10 = 0$ by completing the square. What is the solution?
Match each step to the correct method for solving the equation $5x^2 + 8x + 1 = 0$ using the quadratic formula.
Match each step to the correct method for solving the equation $5x^2 + 8x + 1 = 0$ using the quadratic formula.
Which expression represents the number $3 + √-4$ rewritten in $a + bi$ form?
Which expression represents the number $3 + √-4$ rewritten in $a + bi$ form?
For the number 18, what is the value of a and b?
For the number 18, what is the value of a and b?
Did Lupita make a mistake in completing the square to solve $0 = x^2 - 6x + 13$?
Did Lupita make a mistake in completing the square to solve $0 = x^2 - 6x + 13$?
Did Marcus make a mistake when using the quadratic formula for $0 = x^2 + 5x + 17$?
Did Marcus make a mistake when using the quadratic formula for $0 = x^2 + 5x + 17$?
Which expression is equivalent to $(2 - √-4) + (1 + √-49)$?
Which expression is equivalent to $(2 - √-4) + (1 + √-49)$?
Which expression is equivalent to $(2 - 7i) - (-3 + 2i)$?
Which expression is equivalent to $(2 - 7i) - (-3 + 2i)$?
Given the complex numbers l and m below, what is l - m? l = 3 + 8i, m = 2 - 14i
Given the complex numbers l and m below, what is l - m? l = 3 + 8i, m = 2 - 14i
Which expression is equivalent to $(2 - 3i) + (2 + 7i)$?
Which expression is equivalent to $(2 - 3i) + (2 + 7i)$?
Which expression or value is equivalent to $(8 + 2i)(8 - 2i)$?
Which expression or value is equivalent to $(8 + 2i)(8 - 2i)$?
Which expression is equivalent to $(4 + 6i)(2 + 8i)$?
Which expression is equivalent to $(4 + 6i)(2 + 8i)$?
Which expression is equivalent to $4i(3 + 9i)$?
Which expression is equivalent to $4i(3 + 9i)$?
Which expression represents the number $2i^4 - 5i^3 + 3i^2 + √-81$ rewritten in $a + bi$ form?
Which expression represents the number $2i^4 - 5i^3 + 3i^2 + √-81$ rewritten in $a + bi$ form?
Which expression represents the number $2 + 5i - i(3 - 6i)$ rewritten in $a + bi$ form?
Which expression represents the number $2 + 5i - i(3 - 6i)$ rewritten in $a + bi$ form?
Which simplifications of the powers of $i$ are correct? Select all that apply.
Which simplifications of the powers of $i$ are correct? Select all that apply.
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Study Notes
Factored Forms
- The expression (x^2 + 13x + 42) factors to ((x + 6)(x + 7)).
- The expression (9x^2 - 24x + 16) factors to ((3x - 4)^2).
Solutions to Equations
- For the equation (x^2 + 3x - 28 = 0), the solutions are (4) and (-7).
- The function (f(x) = x^2 + 8x + 16) has a zero at (-4) only.
- The solutions for ((x + 16)^2 = 28) are (x = -16 ± 2\sqrt{7}).
Completing the Square
- Solving (x^2 + 3x - 10 = 0) by completing the square results in the solutions (2) and (-5).
Quadratic Formula Steps
- The quadratic equation (5x^2 + 8x + 1 = 0) can be solved using the formula, with steps clearly outlined showing each calculation.
Complex Numbers
- The expression (3 + \sqrt{-4}) rewritten in (a + bi) form is (3 + 2i).
- For the number (18), (a = 18) and (b = 0).
- The equivalent expression for ((2 - \sqrt{-4}) + (1 + \sqrt{-49})) is (3 + 5i).
- The equivalent expression for ((2 - 7i) - (-3 + 2i)) is (5 - 9i).
- Calculating (l - m) where (l = 3 + 8i) and (m = 2 - 14i), yields (1 + 22i).
Operations with Complex Numbers
- The expression ( (2 - 3i) + (2 + 7i) ) is equivalent to (4 + 4i).
- The expression ((8 + 2i)(8 - 2i)) simplifies to (68).
- The expression ((4 + 6i)(2 + 8i)) simplifies to (-40 + 44i).
- The expression (4i(3 + 9i)) simplifies to (-36 + 12i).
Simplifications of Powers of i
- The correct simplifications include (i^7 = -i) and (i^{16} = 1).
Final Forms of Complex Expressions
- The expression (2i^4 - 5i^3 + 3i^2 + \sqrt{-81}) in (a + bi) form is (-1 + 14i).
- The expression (2 + 5i - i(3 - 6i)) in (a + bi) form is (-4 + 2i`.
Miscellaneous Notes
- Mistakes can occur when applying procedures; note that Lupita should have added (9) in her solution process, and Marcus incorrectly used (2(5)) instead of (2(1)) in the quadratic formula.
- Ensure to double-check all calculations and logic when solving equations.
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