Competition · AMC preparation · step 4 of 4

AMC 10 · 2020A · #14

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

vieta-formulaspolynomial-factoringsymmetric-polynomialsfraction-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: vieta-formulaspolynomial-factoring

📏 Long solution 💡 3 insights

Problem

Real numbers x and y satisfy x + y = 4 and xy = -2. Compute the value of x +

\frac{x³}{y²}

\frac{y³}{x²}

+ y.

Pick an answer.

(A)

360

(B)

400

(C)

420

(D)

440

(E)

480

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): the target E has four terms that don't obviously combine. Group them as (x + y³/x²) + (x³/y² + y) = (x³ + y³)/x² + (x³ + y³)/y², then factor out x³ + y³ to get E = (x³ + y³)(1/x² + 1/y²) = (x³ + y³)(x² + y²)/(x² y²). Tool #13 (Algebra): compute the symmetric sums x² + y², x³ + y³, x² y² from x + y = 4, xy = -2. Tool #3 (Eliminate): match the final number against the five choices.

1STEP 1

Group the four terms

Rewrite x = $\frac{x³}{x²}$ and y = $\frac{y³}{y²}$ so all four terms share denominators, then regroup: E = $\frac{x³ + y³}{x²}$ + $\frac{x³ + y³}{y²}$ .

E = (x³ + y³)/x² + (x³ + y³)/y²

2STEP 2

Combine over one denominator

Factor out the common x³ + y³ and combine the fractions over x² y²: E = $\frac{(x³ + y³)(x² + y²)}{x² y²}$ .

E = (x³ + y³)(x² + y²)/(x² y²)

3STEP 3

Compute the sum of squares

From (x + y)² = x² + 2xy + y²: x² + y² = 4² - 2(-2) = 16 + 4 = 20.

x² + y² = (x+y)² - 2xy = 16 + 4 = 20

4STEP 4

Compute the sum of cubes

From (x+y)³ = (x³ + y³) + 3xy(x+y): x³ + y³ = 4³ - 3(-2)(4) = 64 + 24 = 88.

x³ + y³ = (x+y)³ - 3xy(x+y) = 64 + 24 = 88

5STEP 5

Compute the product squared

Square the product: x² y² = (xy)² = (-2)² = 4.

x² y² = (xy)² = 4

6STEP 6

Substitute and evaluate

Substitute the three pieces: E = $\frac{88 · 20}{4}$ = $\frac{1760}{4}$ = 440.

E = (88 · 20)/4 = 22 · 20 = 440

7STEP 7

Match to the choices

Only choice (D) equals 440; dropping the x² + y² factor or a sign slip gives the smaller options.

E = 440 → (D)

Answer

440

The two roots of t² - 4t - 2 = 0 are x = 2 + √(6), y = 2 - √(6) (since x + y = 4, xy = -2). Then x ≈ 4.449, y ≈ -0.449, so

\frac{x³}{y²}

≈

\frac{88.0}{0.202}

≈ 436 and

\frac{y³}{x²}

≈

\frac{-0.091}{19.79}

≈ -0.005. Adding x + y = 4 gives E ≈ 440. ✓ Matches.

💡Key takeaway

This AMC 10 problem only needs Grade 7 algebra you already know! Rewrite x = $\frac{x³}{x²}$ and y = $\frac{y³}{y²}$ so the four terms factor as E = $\frac{(x³ + y³)(x² + y²)}{x² y²}$ . The identities x² + y² = (x+y)² - 2xy = 20 and x³ + y³ = (x+y)³ - 3xy(x+y) = 88 and x² y² = 4 are all you need. Plug in: E = $\frac{88 · 20}{4}$ = 440, answer (D).

Group the four terms
Combine over one denominator
Compute the sum of squares
Compute the sum of cubes
Compute the product squared
Substitute and evaluate
Match to the choices

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