Questions and Answers
What is the number of real solutions for the equation $x^2 + 9x + 7 = 0$?
Use $b^2 - 4ac$
What is the number of real solutions for the equation $8x^2 - 11x = -3$?
Use $b^2 - 4ac$
What is the number of real solutions for the equation $x^2 = -7x + 7$?
Use $b^2 - 4ac$
What is the number of real solutions for the equation $-4x^2 - 4 = 8x$?
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What is the expression in factored form for $2x^2 + 16x + 30$?
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What is the expression in factored form for $-4x^2 + 8x + 32$?
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What is the expression in factored form for $3x^2 + 26x + 35$?
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What is the expression in factored form for $9x^2 - 18x + 9$?
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What is the expression in factored form for $x^2 - 64$?
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What are the solutions of the quadratic equation $x^2 - 9x + 18 = 0$?
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What are the solutions of the quadratic equation $3x^2 + 25x + 42 = 0$?
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What are the solutions of the quadratic equation $4x^2 - 18x + 20 = 0$?
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Solve by using tables. Give each answer to at most two decimal places for the equation $2x^2 + 5x - 3 = 0$.
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Solve by using tables. Give each answer to at most two decimal places for the equation $-7x^2 - 2 = -10x$.
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What is the solution of the equation $3x^2 = 21$?
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What is the solution of the equation $108x^2 = 147$?
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Solve the equation $x^2 + 18x + 81 = 25$.
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Solve the equation $x^2 - 8x + 16 = 16$.
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Solve the equation $0.125r - 0.0625 + 0.25r = 0.25 + r$.
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Solve the equation $-5y - 9 = - (y - 1)$.
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Solve the quadratic equation by completing the square for $x^2 + 10x + 14 = 0$.
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Solve the quadratic equation by completing the square for $-3x^2 + 7x = -5$.
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Rewrite the equation in vertex form. Name the vertex and y-intercept for $y = rac{3}{5}x^2 + 30x + 382$.
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Use the quadratic formula to solve the equation $-2x^2 - 5x + 5 = 0$.
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Use the quadratic formula to solve the equation $2x^2 + x - 4 = 0$.
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Use the quadratic formula to solve the equation $-4x^2 + x = -4$.
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Simplify the number using the imaginary unit i for $ ext{√}(-144)$.
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Simplify the number using the imaginary unit i for $ ext{√}(-360)$.
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Simplify the expression $(-1 + 6i) + (-4 + 2i)$.
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Simplify the expression $(1 + 2i) - (5 + 3i)$.
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Simplify the expression $(-5i)(6i)$.
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Simplify the expression $(6 - 4i)(-1 + 6i)$.
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Simplify the expression $ rac{-1 + 3i}{4 - i}$.
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Simplify the expression $ rac{-5 + 4i}{-6i}$.
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What are the solutions for $9x^2 + 16 = 0$?
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What are the solutions for $½x^2 - x + 5 = 0$?
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Which algebraic expression models the given word phrase: 40 fewer than a number t?
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Which algebraic expression models the total amount saved starting with $15.00 and saving $8.00 each week?
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Write an algebraic model for the cost of clothes with $30$ for sweaters, $19$ for t-shirts, and $18$ for pants.
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Is the relation a function for the set {(15, 11), (5, 1), (4, 11), (9, 9), (5, 5)}?
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What is the slope of the line that passes through the points (-12, 12) and (2, 4)?
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What is the slope of the line that passes through the points (7, -9) and (7, -1)?
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What is an equation of the line in slope-intercept form with slope $m = rac{1}{2}$ and y-intercept (0, -2)?
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What is the graph of the equation $3x - y = 3$?
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What is the graph of the equation $-5x + 5y = 4$?
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Write an equation of the line in point-slope form that passes through the points (-9, 4) and (13, -7).
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What is an equation of the line in point-slope form that passes through the point (9, 6) and has slope $1/3$?
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Study Notes
Real Solutions of Quadratic Equations
- The number of real solutions can be found using the discriminant formula ( b^2 - 4ac ).
- Examples include:
- For ( x^2 + 9x + 7 = 0 ), calculate ( b^2 - 4ac ).
- For ( 8x^2 - 11x + 3 = 0 ), apply the same method.
- (-4x^2 - 4 = 8x) also requires this calculation.
Factored Form of Quadratic Expressions
- Quadratic expressions can be factored:
- Example: ( 2x^2 + 16x + 30 ) factors to ( 2(x+3)(x+5) ).
- Additional examples need solving or factoring like ( -4x^2 + 8x + 32 ).
Solutions to Quadratic Equations
- Quadratic equations can yield specific solutions or require techniques:
- Use the quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Example equations: ( x^2 - 9x + 18 = 0 ) or ( 3x^2 + 25x + 42 = 0 ).
Solving Quadratic by Various Methods
- Equations can be solved through different methods such as:
- Completing the square for ( x^2 + 10x + 14 = 0 ).
- Using numerical tables for equations like ( 2x^2 + 5x - 3 = 0 ).
Using Imaginary Units
- Simplification involving the imaginary unit ( i ) is essential for non-real results:
- For example, ( \sqrt{-144} = 12i ) and ( \sqrt{-360} = 6i\sqrt{10} ).
Simplifying Expressions with Complex Numbers
- Operations on complex numbers need proper simplifications:
- For ( (-1 + 6i) + (-4 + 2i) ), combine like terms.
- Products like ( (-5i)(6i) ) need careful handling of ( i^2 = -1 ).
Evaluating Algebraic Models and Functions
- Algebraic expressions from word problems should be established:
- Modeling savings like starting amount minus weekly savings.
- Evaluating costs based on quantities and prices of items, e.g., sweaters, t-shirts, pants.
Slope and Linear Equations
- Understanding the slope from given points helps in identifying line characteristics:
- Points like ( (-12,12) ) and ( (2,4) ) provide slope calculations.
- Point-slope forms can be derived from known points and slopes.
Graphical Representation of Equations
- Graphs visualize linear equations like ( 3x - y = 3 ) or ( -5x + 5y = 4 ).
- Converting standard forms to slope-intercept form allows for easier graphing.
Characteristics of Functions
- Determining if a relation is a function, as in the set ( {(15, 11), (5, 1), (4,11), (9,9),(5,5)} ), where no input maps to multiple outputs is critical.
These study notes encapsulate key concepts and techniques needed for mastering Algebra 2, particularly in the context of solving and understanding quadratic equations, functions, and expressions.
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Description
This quiz covers the fundamentals of quadratic equations, including methods for finding real solutions using the discriminant and the quadratic formula. It also explores factoring quadratic expressions and solving them through various techniques. Test your knowledge with examples and gain a deeper understanding of quadratic functions.
