Math 2 Quiz: Choosing Quadratic Solution Methods
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Choosing Quadratic Solution MethodsQuestion 1 of 18

A company determines that their weekly revenue R (in thousands of dollars) from producing x thousand units is R(x) = 8x - x².

To find the production levels that yield exactly $15,000 in weekly revenue, which method would encounter the most computational difficulty?

Factoring x² - 8x + 15 = 0 by finding two numbers that multiply to 15 and add to -8
Completing the square on x² - 8x + 15 = 0 by adding and subtracting the square of half the linear coefficient
Quadratic formula with a = 1, b = -8, c = 15, calculating the discriminant as 64 - 60 = 4
Graphing y = 8x - x² and y = 15 to find intersections, requiring precise scale and accurate plotting
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Math 2 Quiz

Math 2 Quiz: Choosing Quadratic Solution Methods

Practice Choosing Quadratic Solution Methods in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Choosing Quadratic Solution Methods, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A company determines that their weekly revenue R (in thousands of dollars) from producing x thousand units is R(x) = 8x - x².

To find the production levels that yield exactly $15,000 in weekly revenue, which method would encounter the most computational difficulty?

  1. Factoring x² - 8x + 15 = 0 by finding two numbers that multiply to 15 and add to -8
  2. Completing the square on x² - 8x + 15 = 0 by adding and subtracting the square of half the linear coefficient
  3. Quadratic formula with a = 1, b = -8, c = 15, calculating the discriminant as 64 - 60 = 4
  4. Graphing y = 8x - x² and y = 15 to find intersections, requiring precise scale and accurate plotting (correct answer)
Explanation: Setting R(x) = 15: 8x - x² = 15, so x² - 8x + 15 = 0. This factors as (x - 3)(x - 5) = 0, giving x = 3 or x = 5 thousand units. Factoring is easiest since 3 and 5 multiply to 15 and add to 8. Completing the square and the quadratic formula also work cleanly. Graphing requires the most care in scaling and plotting to get precise intersection points.

Question 2

A rocket is launched from ground level. Its height h (in feet) after t seconds is given by h(t) = -16t² + 128t.

To determine the total flight time (when the rocket returns to ground level), which method is most efficient?

  1. Quadratic formula on -16t² + 128t = 0, treating this as a standard quadratic equation
  2. Factoring out the common factor: t(-16t + 128) = 0, then solving each factor separately (correct answer)
  3. Completing the square to rewrite as -16(t - 4)² + 256 = 0, then solving for t
  4. Using the vertex formula to find the maximum height time, then doubling it due to symmetry
Explanation: Setting h(t) = 0: -16t² + 128t = 0. Factoring: t(-16t + 128) = 0. This gives t = 0 (launch time) or -16t + 128 = 0, so t = 8 seconds (landing time). Factoring is most efficient because the common factor t immediately reveals both the launch and landing times. The rocket is in flight for 8 seconds total.

Question 3

A suspension bridge cable hangs in a parabolic shape. The height y (in feet) above the roadway at a distance x (in feet) from the center is given by y = 0.01x² + 10.

To find the horizontal distances where the cable is 35 feet above the roadway, which solution approach is most appropriate?

  1. Factoring 0.01x² - 25 = 0 after rearranging, then applying the difference of squares pattern (correct answer)
  2. Completing the square on 0.01x² + 10 = 35 to rewrite in vertex form with integer coefficients
  3. Quadratic formula on 0.01x² + 0x + (10-35) = 0 with careful decimal arithmetic throughout
  4. Graphing y = 0.01x² + 10 and y = 35 simultaneously to find intersection points using technology
Explanation: Setting y = 35: 0.01x² + 10 = 35, so 0.01x² = 25, giving x² = 2500. This factors as x² - 2500 = 0, which is a difference of squares: (x - 50)(x + 50) = 0. Therefore x = ±50 feet from center. Factoring using difference of squares is most efficient here since the decimal coefficient leads to a perfect square when isolated.

Question 4

A company's revenue R (in dollars) from selling x items is given by R(x) = 25x - 0.5x².

To find how many items must be sold to generate exactly $312.50 in revenue, which approach is most practical?

  1. Quadratic formula on 0.5x² - 25x + 312.5 = 0, multiplying through by 2 to eliminate decimals first (correct answer)
  2. Completing the square on 25x - 0.5x² = 312.5 by factoring out the leading coefficient
  3. Factoring 0.5x² - 25x + 312.5 = 0 directly by finding two numbers that multiply to give the product ac
  4. Graphing both y = 25x - 0.5x² and y = 312.5 to find their intersection points visually
Explanation: Setting 25x - 0.5x² = 312.5 gives 0.5x² - 25x + 312.5 = 0. Multiplying by 2: x² - 50x + 625 = 0. Using the quadratic formula: x = (50 ± √(2500 - 2500))/2 = 50/2 = 25. The discriminant is 0, so there's exactly one solution: x = 25 items. The quadratic formula is most systematic here, especially after eliminating decimals.

Question 5

A water balloon is thrown upward from a height of 5 feet with initial velocity 32 ft/s. Its height is modeled by h(t) = -16t² + 32t + 5.

A student wants to find when the balloon is at a height of 21 feet. Which method would be least efficient for this specific problem?

  1. Completing the square, since the resulting perfect square trinomial leads to clean calculations
  2. Quadratic formula, since -16t² + 32t - 16 = 0 simplifies to t² - 2t + 1 = 0 after factoring
  3. Factoring, since t² - 2t + 1 = (t - 1)² = 0 gives an immediate answer
  4. Graphing both y = -16t² + 32t + 5 and y = 21 to find intersection points using technology (correct answer)
Explanation: Setting h(t) = 21: -16t² + 32t + 5 = 21, so -16t² + 32t - 16 = 0. Dividing by -16: t² - 2t + 1 = 0. This factors as (t - 1)² = 0, giving t = 1. Both factoring and completing the square are efficient since this is a perfect square trinomial. The quadratic formula also works well. Graphing is least efficient because it requires technology and gives approximate rather than exact answers.

Question 6

A ball is dropped from a building. Its height above ground is given by h(t) = -16t² + 144, where h is in feet and t is in seconds.

To determine when the ball hits the ground, which method is most efficient given the structure of this equation?

  1. Quadratic formula, applying it directly to -16t² + 0t + 144 = 0 with a = -16, b = 0, c = 144
  2. Factoring out the GCD first: -16(t² - 9) = 0, then using difference of squares (correct answer)
  3. Completing the square by rewriting as -16(t² - 0t) + 144 = 0 and adjusting the constant
  4. Graphing the function to visually determine where the parabola intersects the t-axis
Explanation: Setting h(t) = 0: -16t² + 144 = 0. Factoring out -16: -16(t² - 9) = 0, so t² - 9 = 0. This is a difference of squares: (t - 3)(t + 3) = 0, giving t = 3 or t = -3. Since time must be positive, t = 3 seconds. This method is most efficient because it avoids the quadratic formula and immediately recognizes the difference of squares pattern.

Question 7

An online retailer finds that their daily profit P (in hundreds of dollars) depends on the number of ads x they run according to P(x) = -x² + 12x - 27.

To determine the number of ads that results in zero profit (break-even points), which method reveals the most insight about the problem structure?

  1. Quadratic formula, since it works universally and gives exact values for any quadratic equation
  2. Completing the square to get P(x) = -(x - 6)² + 9, revealing the vertex and axis of symmetry
  3. Factoring as P(x) = -(x - 3)(x - 9), immediately showing the break-even points and their relationship (correct answer)
  4. Graphing to visualize the parabola and identify all key features including intercepts and maximum
Explanation: Setting P(x) = 0: -x² + 12x - 27 = 0, or x² - 12x + 27 = 0. This factors as (x - 3)(x - 9) = 0, giving break-even at x = 3 and x = 9 ads. Factoring immediately reveals that break-even occurs at 3 and 9 ads, and shows these values are symmetric about x = 6. This provides the most insight into the problem structure, showing the relationship between the break-even points.

Question 8

An architect designs a rectangular garden where the length is 8 feet more than the width, and the total area must be exactly 240 square feet.

After setting up the equation w(w + 8) = 240, which solution method would be most appropriate for finding the width w?

  1. Completing the square, since expanding gives w² + 8w - 240 = 0 with coefficient 8 that's easily halved
  2. Factoring by grouping, since 240 has many factor pairs that might work with the linear term
  3. Quadratic formula, since the discriminant 64 + 960 = 1024 = 32² gives integer solutions (correct answer)
  4. Graphing y = w² + 8w - 240 and finding where it crosses the w-axis using technology
Explanation: The equation w² + 8w - 240 = 0 has discriminant b² - 4ac = 64 + 960 = 1024 = 32². This gives w = (-8 ± 32)/2, so w = 12 or w = -20. Since width must be positive, w = 12. While factoring might work, finding factors of 240 that differ by 8 requires trial and error. The quadratic formula is more systematic and the perfect square discriminant makes calculation clean.

Question 9

A manufacturer's cost C (in dollars) to produce x units is given by C(x) = x² - 60x + 1400.

To find the production level that minimizes cost, a student considers four approaches. Which method most directly reveals both the minimum cost and the optimal production level?

  1. Using vertex formula x = -b/(2a) = 30, then substituting to find C(30)
  2. Completing the square: C(x) = (x - 30)² + 500 (correct answer)
  3. Taking derivative C'(x) = 2x - 60, setting to zero, solving for x
  4. Factoring the quadratic and finding the midpoint between roots
Explanation: Completing the square gives C(x) = x² - 60x + 1400 = (x - 30)² - 900 + 1400 = (x - 30)² + 500. This directly shows the minimum cost is $500 when x = 30 units. While the vertex formula works, it requires a separate substitution step. Option C uses calculus. Option D won't work since this quadratic (when set equal to zero) has no real roots.

Question 10

A stone is thrown upward from a 48-foot building with initial velocity 32 ft/s. Its height is h(t)=−16t2+32t+48h(t) = -16t^2 + 32t + 48. A safety coordinator needs to know when the stone will be exactly 64 feet high to clear people from that level. The coordinator needs a method that clearly shows whether this height is reached once or twice during the flight. Which approach best serves this need?

  1. Factoring, because −16t2+32t−16=0-16t^2 + 32t - 16 = 0 factors as −16(t−1)2=0-16(t-1)^2 = 0 showing one solution
  2. Quadratic formula, because the discriminant indicates the number of real solutions directly (correct answer)
  3. Completing the square, because the vertex form shows the maximum height relative to 64 feet
  4. Graphing, because the visual intersection of y=64y = 64 with the parabola shows intersection multiplicity
Explanation: When you need to determine how many times a projectile reaches a specific height, you're solving a quadratic equation and need to know the number of real solutions. This is exactly what the discriminant tells you directly. To find when the stone reaches 64 feet, set h(t)=64h(t) = 64: −16t2+32t+48=64-16t^2 + 32t + 48 = 64, which simplifies to −16t2+32t−16=0-16t^2 + 32t - 16 = 0. Using the quadratic formula t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, the discriminant is b2−4ac=322−4(−16)(−16)=1024−1024=0b^2 - 4ac = 32^2 - 4(-16)(-16) = 1024 - 1024 = 0. Since the discriminant equals zero, there's exactly one real solution, meaning the stone reaches 64 feet at exactly one moment during its flight. Choice A attempts factoring but makes an error. The correct factorization of −16t2+32t−16=0-16t^2 + 32t - 16 = 0 is −16(t2−2t+1)=−16(t−1)2=0-16(t^2 - 2t + 1) = -16(t-1)^2 = 0, which does show one solution, but the discriminant method is more systematic and reliable. Choice C's completing the square approach can work but requires additional steps to interpret whether the height is reached once or twice, making it less direct for this specific question. Choice D suggests graphing, but determining intersection multiplicity visually can be imprecise, especially when dealing with tangent points rather than clear crossings. Remember: when a problem asks specifically about the number of solutions to a quadratic equation, check the discriminant first. If b2−4ac>0b^2 - 4ac > 0, there are two real solutions; if it equals zero, there's one; if negative, there are none.

Question 11

A rectangular garden has dimensions where the length is 3 feet more than twice the width. If the total area must be 119 square feet, which solution method would be most efficient for finding the width, considering that the width must be a positive integer?

  1. Factoring, since the resulting quadratic 2w2+3w−119=02w^2 + 3w - 119 = 0 has rational roots
  2. Quadratic formula, because the discriminant 9+4(2)(119)9 + 4(2)(119) does not yield perfect squares
  3. Completing the square, because the coefficient 2 can be factored out easily
  4. Factoring, since 2w2+3w−119=02w^2 + 3w - 119 = 0 can be written as (2w+17)(w−7)=0(2w + 17)(w - 7) = 0 (correct answer)
Explanation: If width = ww, then length = 2w+32w + 3. Area equation: w(2w+3)=119w(2w + 3) = 119, so 2w2+3w−119=02w^2 + 3w - 119 = 0. This factors as (2w+17)(w−7)=0(2w + 17)(w - 7) = 0, giving w=7w = 7 (since width must be positive). Choice A is incorrect because while the equation has rational roots, it doesn't factor as simply as stated. Choice B is wrong because the discriminant 9+952=961=3129 + 952 = 961 = 31^2 is a perfect square. Choice C is less efficient than the direct factoring approach.

Question 12

A bridge's arch follows the equation y=−0.02x2+1.6x−12y = -0.02x^2 + 1.6x - 12, where xx is the horizontal distance in feet and yy is the height in feet. Engineers need to find where the arch intersects the water level at y=20y = 20 feet to place support pillars. Given that precise measurements are critical and the solutions are not likely to be integers, which method is most suitable?

  1. Factoring, because rearranging gives 0.02x2−1.6x+32=00.02x^2 - 1.6x + 32 = 0 with rational coefficients
  2. Completing the square after multiplying by 50 to eliminate decimals completely
  3. Quadratic formula, because the decimal coefficients and non-integer solutions require precise calculation (correct answer)
  4. Graphing with technology, because the intersection points can be found using trace functions accurately
Explanation: Setting y=20y = 20: −0.02x2+1.6x−12=20-0.02x^2 + 1.6x - 12 = 20, so −0.02x2+1.6x−32=0-0.02x^2 + 1.6x - 32 = 0. With decimal coefficients and the need for precise non-integer solutions, the quadratic formula is most appropriate: x=−1.6±1.62−4(−0.02)(−32)2(−0.02)x = \frac{-1.6 \pm \sqrt{1.6^2 - 4(-0.02)(-32)}}{2(-0.02)}. Choice A is incorrect because the equation doesn't factor nicely. Choice B adds unnecessary complexity by requiring multiplication. Choice D may introduce rounding errors that compromise the precision needed for engineering work.

Question 13

A water fountain shoots water in a parabolic arc described by h=−x2+8x−7h = -x^2 + 8x - 7, where hh is height in feet and xx is horizontal distance in feet. A designer needs to place decorative elements at ground level (h=0h = 0) where the water lands, and also wants to understand the fountain's range and symmetry for aesthetic planning. Which solution method provides the most comprehensive information for design purposes?

  1. Factoring, because x2−8x+7=(x−1)(x−7)x^2 - 8x + 7 = (x-1)(x-7) directly gives landing points x=1x = 1 and x=7x = 7
  2. Quadratic formula, because it systematically finds the zeros and provides exact coordinates for decoration placement
  3. Graphing with multiple viewing windows, because it shows the complete parabolic shape and design context visually
  4. Completing the square, because h=−(x−4)2+9h = -(x-4)^2 + 9 shows the range, maximum height, and symmetry axis (correct answer)
Explanation: When you encounter parabolic motion problems that ask for "comprehensive information" or "design purposes," you need to think beyond just finding zeros. The question is testing which form of a quadratic equation reveals the most useful geometric properties. Completing the square transforms h=−x2+8x−7h = -x^2 + 8x - 7 into vertex form: h=−(x−4)2+9h = -(x-4)^2 + 9. This immediately reveals that the vertex is at (4,9)(4, 9), meaning the water reaches maximum height of 9 feet at horizontal distance 4 feet. The axis of symmetry is x=4x = 4, which tells the designer the fountain's centerline for symmetric decoration placement. The range spans from the landing points (found by setting h=0h = 0) to the maximum height. Option A is incomplete because factoring only gives you the landing points x=1x = 1 and x=7x = 7, but provides no information about maximum height, vertex location, or the axis of symmetry that's crucial for aesthetic planning. Option B, while the quadratic formula does find the zeros systematically, it doesn't directly reveal the geometric properties like vertex and symmetry axis that are essential for comprehensive design planning. Option C might seem appealing since graphing shows visual context, but the question asks which solution method provides the most information, not which visualization tool is best. For parabola problems involving design, motion, or optimization, always consider completing the square when you need comprehensive geometric information. Vertex form a(x−h)2+ka(x-h)^2 + k immediately shows you the vertex (h,k)(h,k), axis of symmetry, and direction of opening.

Question 14

A company's revenue model is R(p)=−50p2+1200pR(p) = -50p^2 + 1200p, where pp is the price per unit in dollars. The marketing team needs to find what price will generate exactly $7000 in revenue, and they suspect there might be two different pricing strategies that achieve this target. However, they want to avoid any computational errors since this affects major business decisions. Which method minimizes calculation errors while confirming the number of solutions?

  1. Completing the square on p2−24p+140=0p^2 - 24p + 140 = 0, because it avoids fraction arithmetic entirely
  2. Factoring after dividing by 50, because p2−24p+140=0p^2 - 24p + 140 = 0 has smaller coefficients to work with (correct answer)
  3. Quadratic formula applied to 50p2−1200p+7000=050p^2 - 1200p + 7000 = 0, because it's systematic and verifiable
  4. Graphing y=−50p2+1200py = -50p^2 + 1200p and y=7000y = 7000, because technology eliminates human calculation errors
Explanation: When you encounter revenue optimization problems with quadratic equations, your first priority should be minimizing computational complexity while maintaining accuracy. Here, you need to solve −50p2+1200p=7000-50p^2 + 1200p = 7000 to find the pricing strategies. The key insight is recognizing that dividing the entire equation by the greatest common factor (50) dramatically simplifies your work. Rearranging to standard form gives 50p2−1200p+7000=050p^2 - 1200p + 7000 = 0, and dividing by 50 yields p2−24p+140=0p^2 - 24p + 140 = 0. These smaller coefficients make every subsequent calculation easier and less error-prone, whether you factor, complete the square, or use the quadratic formula. This approach gives you the cleanest path to finding both solutions. Choice A incorrectly claims completing the square avoids fractions entirely—you'll still encounter fractions when finding (p−12)2=4\left(p - 12\right)^2 = 4, leading to p=12±2p = 12 \pm 2. Choice C suggests using the quadratic formula on the original equation, but working with coefficients like 1200 and 7000 increases arithmetic errors significantly. Choice D recommends graphing technology, but the question specifically asks for a method that minimizes calculation errors through mathematical technique, not by avoiding calculation altogether. The correct answer is B because simplifying coefficients first creates the most manageable equation regardless of which solution method you choose afterward. Remember: when facing quadratic equations with large coefficients, always check if you can factor out a common divisor first—it's often the difference between clean arithmetic and computational chaos.

Question 15

A farmer wants to create a rectangular pen against an existing fence, using 200 feet of new fencing for three sides. The area function is A(w)=w(200−2w)A(w) = w(200-2w) where ww is the width. To find the width that gives exactly 4800 square feet, and considering this problem requires understanding the relationship between width and maximum possible area, which method is most insightful?

  1. Factoring, because 2w2−200w+4800=02w^2 - 200w + 4800 = 0 simplifies to w2−100w+2400=0w^2 - 100w + 2400 = 0
  2. Quadratic formula, because it handles the large coefficients in 2w2−200w+4800=02w^2 - 200w + 4800 = 0 systematically
  3. Completing the square, because it reveals how the desired area compares to the maximum possible area (correct answer)
  4. Graphing, because the parabola shows both the area function behavior and specific intersection points
Explanation: Setting w(200−2w)=4800w(200-2w) = 4800 gives 2w2−200w+4800=02w^2 - 200w + 4800 = 0, or w2−100w+2400=0w^2 - 100w + 2400 = 0. Completing the square: (w−50)2−2500+2400=0(w-50)^2 - 2500 + 2400 = 0, so (w−50)2=100(w-50)^2 = 100. This shows the maximum area occurs at w=50w = 50 (vertex), and our desired area of 4800 is 100 square feet less than the maximum of 5000, giving solutions w=40w = 40 and w=60w = 60. Choice A works but doesn't provide insight about the maximum. Choice B is computational but not insightful. Choice D doesn't emphasize the relationship between desired and maximum areas.

Question 16

A projectile's height hh (in meters) at time tt seconds is given by h(t)=−4.9t2+24.5t+1.5h(t) = -4.9t^2 + 24.5t + 1.5. An engineer needs to find when the projectile reaches its maximum height to optimize a catching mechanism. Which method is most appropriate for this specific requirement?

  1. Quadratic formula, because it directly gives the time values when height equals zero
  2. Factoring, because the equation can be rewritten with integer coefficients throughout
  3. Completing the square, because it reveals the vertex form showing maximum height and time (correct answer)
  4. Graphing, because the vertex can be estimated accurately from the parabola's visual peak
Explanation: To find maximum height, we need the vertex of the parabola. Completing the square transforms the equation to vertex form h(t)=−4.9(t−2.5)2+32.125h(t) = -4.9(t - 2.5)^2 + 32.125, directly showing the maximum occurs at t=2.5t = 2.5 seconds with height 32.125 meters. Choice A finds zeros, not the maximum. Choice B is incorrect because the decimal coefficients don't factor nicely with integers. Choice D is less precise than the algebraic method for engineering applications requiring exact values.

Question 17

A basketball is shot from ground level with an initial height of 6 feet. Its height hh (in feet) after tt seconds is modeled by h(t)=−16t2+32t+6h(t) = -16t^2 + 32t + 6. A coach wants to determine when the ball will be at exactly 22 feet above ground, but needs to choose the most efficient solution method given that this height occurs twice during the ball's flight. Which method would be most appropriate and why?

  1. Factoring, because the equation −16t2+32t−16=0-16t^2 + 32t - 16 = 0 has integer coefficients that factor cleanly (correct answer)
  2. Completing the square, because the coefficient of t2t^2 creates perfect square trinomials easily
  3. Quadratic formula, because after rearranging, the discriminant will not yield integer solutions
  4. Graphing, because finding intersection points is more accurate than algebraic methods for this context
Explanation: Setting h(t)=22h(t) = 22: −16t2+32t+6=22-16t^2 + 32t + 6 = 22, so −16t2+32t−16=0-16t^2 + 32t - 16 = 0. Dividing by −16-16: t2−2t+1=0t^2 - 2t + 1 = 0, which factors as (t−1)2=0(t-1)^2 = 0. The factoring method is most efficient here. Choice B is wrong because completing the square is unnecessarily complex when factoring works easily. Choice C is incorrect because the discriminant is 0, giving the integer solution t=1t = 1. Choice D is wrong because algebraic methods are more precise than graphing for exact values.

Question 18

A company's profit PP (in thousands of dollars) is modeled by P(x)=−2x2+16x−24P(x) = -2x^2 + 16x - 24, where xx is the number of units produced (in hundreds). The manager needs to determine the break-even points where profit equals zero, but the accounting software can only handle exact fractional answers. Which solution method should be used?

  1. Factoring, because −2x2+16x−24=−2(x−2)(x−6)-2x^2 + 16x - 24 = -2(x-2)(x-6) gives exact integer solutions (correct answer)
  2. Completing the square, because it provides the vertex form needed for profit analysis
  3. Quadratic formula, because the discriminant ensures precise fractional results for accounting
  4. Graphing, because visual intersection with the x-axis shows break-even points clearly
Explanation: For break-even, set P(x)=0P(x) = 0: −2x2+16x−24=0-2x^2 + 16x - 24 = 0. Factoring out −2-2: −2(x2−8x+12)=0-2(x^2 - 8x + 12) = 0, so x2−8x+12=0x^2 - 8x + 12 = 0, which factors as (x−2)(x−6)=0(x-2)(x-6) = 0. This gives exact solutions x=2x = 2 and x=6x = 6. Choice B finds the vertex, not zeros. Choice C is unnecessarily complex when factoring gives exact integer solutions. Choice D provides approximations, not the exact values required for accounting.