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Mathbound · Contest practice

Quadratics worksheet

Mixed difficulty

Name

Date

1.
If $r$ and $s$ are roots of $x^2+8x+10=0$ , find $r^3+s^3$ (without solving).
- A.272
- B.262
- C.280
- D.80
- E.512
2.
If $x+\dfrac1x=5$ , then $x^2+\dfrac1{x^2}=$
- A.25
- B.23
- C.27
- D.21
- E.24
3.
If $r$ and $s$ are roots of $x^2+6x+24=0$ , find $r^3+s^3$ (without solving).
- A.216
- B.-216
- C.-240
- D.-210
- E.144
4.
If $r$ and $s$ are roots of $x^2+7x+8=0$ , find $r^3+s^3$ (without solving).
- A.182
- B.56
- C.343
- D.175
- E.167
5.
If $r$ and $s$ are roots of $x^2+9x+14=0$ , find $r^3+s^3$ (without solving).
- A.126
- B.729
- C.351
- D.337
- E.360
6.
If $r$ and $s$ are roots of $x^2+9x+20=0$ , find $r^3+s^3$ (without solving).
- A.169
- B.198
- C.180
- D.729
- E.189
7.
What is the maximum product of positive integers that sum to $27$ ?
- A.19686
- B.19684
- C.8192
- D.19681
- E.19683
8.
What is the maximum product of positive integers that sum to $20$ ?
- A.1456
- B.1459
- C.1024
- D.729
- E.1458
9.
What is the maximum product of positive integers that sum to $37$ ?
- A.708586
- B.708589
- C.262144
- D.531441
- E.708588
10.
If $r$ and $s$ are roots of $x^2+7x+22=0$ , find $r^3+s^3$ (without solving).
- A.-141
- B.-112
- C.154
- D.343
- E.-119

Answer key · worked solutions

1. A. 272
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=8$ , $rs=10$ . $272$ .
2. B. 23
Square both sides. $25-2=23$ . $23$ .
3. B. -216
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=6$ , $rs=24$ . $-216$ .
4. D. 175
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=7$ , $rs=8$ . $175$ .
5. C. 351
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=9$ , $rs=14$ . $351$ .
6. E. 189
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=9$ , $rs=20$ . $189$ .
7. E. 19683
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $27$ optimally. $19683$ .
8. E. 1458
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $20$ optimally. $1458$ .
9. E. 708588
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $37$ optimally. $708588$ .
10. E. -119
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=7$ , $rs=22$ . $-119$ .

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