Questions and Answers
What is the vertex form of the equation y = -6x² - 24x - 31?
y = -6(x + 2)² - 7
What are the coordinates of the vertex for the equation y = -6x² - 24x - 31?
(-2, -7)
Use the quadratic formula to solve the equation 6x² - 24x - 30 = 0.
5, -1
What are the exact solutions for the equation 2(x + 4)² - 48 = 0?
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Simplify the expression (2 + 5i)/(5 + 4i).
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What is the solution set for the quadratic inequality x² - 2x ≥ 24?
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Write the quadratic equation in vertex form for y = x² + 14x + 4.
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Which inequality is best represented by the graph?
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Use factoring and the Zero-Product Property to find the zeros of the quadratic function f(x) = x² - 7x - 8.
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F(x) = -1 + 9x² + 7x is a quadratic function.
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Find the value of x to the nearest hundredth.
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Solve the quadratic inequality x² - 4x + 3 < 0 and graph the solution on a number line.
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Factor the quadratic expression x² - 11x + 30.
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What is the quadratic equation that best fits the baseball's trajectory described?
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Write the function in the form f(x) = ax² + bx + c and identify a, b, and c for f(x) = (5x - 4)(5x + 3).
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Find a quadratic equation that models the turnstile data. What is the equation?
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Solve the equation by completing the square for x² - 2x - 24 = 0. What are the solutions?
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What equation represents the value remaining in Hans's account after x visits?
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What is the value remaining in Hans's account after 11 visits?
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Solve the proportion (x−8)/8 = 4/5.
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Solve for x in the equation x - 7 = 1.
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Solve for b in the formula B = 7a²b.
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Which equation best describes the graph?
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Solve for x in the equation 3x - 3 = x - 9.
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What inequality has the solution shown in the graph?
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What inequality describes the graph?
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What is the equation of the line of best fit for the given data?
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Write a proportion that models the statement 10 is to 5 as 16 is to 8.
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Write an equation in slope-intercept form for a line that passes through points (4,-7) and (-2,-10).
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Which is the equation of the line with slope -4 and y-intercept 5?
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Identify the inequality which results in an acceptable number of golf balls in each bag.
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What will the population of Abbiville be in 2014 if growth remains constant?
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Match the following expressions with their evaluations:
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If x = 78 when y = 130, find x when y = 190.
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Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
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What is the approximate average cost per T-shirt when the company makes 10 shirts?
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What is the reduced row-echelon form of the given system of equations?
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Find the maximum and minimum values of C = 3x + 5y given the constraints.
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Study Notes
Algebra Concepts and Solutions
- For athletes, an account balance can be modeled as V = 205 - 10x, where x represents visits. After 11 visits, the balance is $95.
- Solved proportion: (x−8)/8 = 4/5 results in x = 14 or 72/5.
- Simple equation solution: x - 7 = 1 leads to x = 8.
- Rearranged equation to isolate b: B = 7a²b becomes b = B/(7a²).
- Linear equation from a graph is represented as y = -2x + 5.
- Solve 3x - 3 = x - 9 to find x = -3.
- The inequality m + 7 ≥ 9 follows from a specific graph solution.
- For x ≤ -5, this describes another graph visually represented.
- The line of best fit for given data yields y = 0.63x + 1.90.
- Proportion of 10 to 5 as 16 to 8 is written as 10/5 = 16/8.
- Using points (4,-7) and (-2,-10), the slope-intercept form is y = 1/2x - 9.
- For points (-6, 5) and (-9, 0), the line is y = 5x/3 + 15.
- The equation for a parallel line through (-5, 4) to y = -4x + 1 is y = -4x - 16.
- Acceptable ranges for golf balls are modeled by the inequality: 365 ≥ 375.
- Estimating Abbiville's population using given data results in 364 inhabitants in 2014.
- The equation for a line with slope -4 and y-intercept 5 can be expressed as 4x + y - 5 = 0 or y = -4x + 5.
- Data from restaurant requests illustrates a positive linear relationship.
- The equation for a perpendicular line passing through (-1, -5) to -8x - 4y = 3 becomes y = -2x - 7.
- An equation with slope -4 through point (3, -6) is y = -4x + 6.
- Direct variation from y = 130 when x = 78 finds x = 114 when y = 190, with a constant k = 0.6.
- The transformation y = f(x-3) involves shifting the graph 3 units to the right.
- The evaluation of (2/5)⁵ results in 32/3125.
- The Commutative Property of Addition is exemplified: 5 + 2 = 2 + 5.
- Evaluating 64 × 4² - 2 × 2² gives 1016.
- Cost function for decorating stuffed animals: C(x) = 5.25x + 44.
- Total cost for 25 decorated animals comes to $175.25. A budget of $227.75 allows decorating 35 animals.
- Function composition: (g o f)(x) = x² - 12x + 34, (f o g)(x) = -x² + 8 for f(x) = 6 - x, g(x) = x² - 2.
- Evaluating (3/8)⁻² gives 64/9.
- From y = 15(x + 2) + 4, the graph translates up 4 units and stretches vertically by a factor of 5.
- Simplification of expression results in -u⁶⁰ from ((-u)³(-u⁹)⁸)/(u⁵)³.
- Squaring (-3/8) produces 9/64.
- The transformation of f(x) = x² by vertical stretching and translations yields g(x) = 3(x - 4)² - 8.
- Average cost function A(x) = (3.75x + 200)/x calculates to $23.75 for 10 shirts.
- Inverses of functions: for y = 7x², the inverse y = ±√(x/7) is not a function.
- A model for customer flow at The Burger Barn estimates 864,000 customers in 1999 from C(t) = t² + 38t + 600.
- Multiplication of functions generates (f × g)(x) = x³ - 3x² - 9x + 27 for f(x) = 9 - x², g(x) = 3 - x.
- Solve the quadratic equation 4x² = 8, yielding solutions ±√2 and approximate values ±1.41.
- Factoring the quadratic equation 2x² + 5x - 3 = 0 provides roots of 1/2 and -3.
- Quadratic fitting results in y = 0.09x² - 0.98x + 4.99 for points (1, 4.1), (5, 2.3), (11, 4.9).
- Solve quadratic inequalities graphically: x² - 2x ≥ 24 yields x ≤ -4 or x ≥ 6.
- Vertex form conversion for y = x² + 14x + 4 results in y = (x + 7)² - 45.
- Simplification of the expression (2 + 5i)/(5 + 4i) gives (30/41) + (17/41)i.
- Identifying solutions for quadratic inequality x² - 4x + 3 < 0 results in 1 < x < 3.
- Zero-finding of the function f(x) = x² - 7x - 8 yields roots 8 and -1.
- Analysis of matrices and equations provides solutions through various methods like elimination and substitution.### Quadratic Equations and Functions
- A quadratic equation takes the form y = ax² + bx + c where a, b, and c are constants.
- Example quadratic equation for baseball's trajectory: y = -0.0383x² + 2.143x + 50.
Identifying Quadratic Functions
- Given expression: f(x) = (5x - 4)(5x + 3) can be written in the standard form.
- Expansion results in: f(x) = 25x² - 5x - 12.
- Coefficients identified: a = 25, b = -5, c = -12.
Modeling Data with Quadratic Equations
- Turnstile data at State Fair can be modeled with the quadratic equation: y = 0.083x² - 0.197x + 7.62.
- Data analysis utilizes graphing calculators to fit quadratic models:
- Enter x values into list L1 and y values into list L2.
- Use Quadratic Regression function to derive the equation.
Solving Quadratic Equations by Completing the Square
- Example equation: x² - 2x - 24 = 0.
- Completing the square provides exact solutions: x = -4, x = 6.
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