What Is A Smart Bird’s Favorite Type of Math? To answer the question, use the values of a, b, and c of each of the following quadratic equations in determining the sum and product of their roots. Write the corresponding letter in the box matching the given sum and product. L x^2 + x = 6 B 12x^2 + x – 6 = 0 G x^2 – 2x – 2 = 0 R 2x^2 + 5x – 25 = 0 A 6x^2 – 5x – 4 = 0 O 3x^2 – 11x – 4 = 0 E 4x – 5 = -4x^2 W 8x^2 + 12x = -1 Answer Box answer sum of roots product of roots 11 3 -3 2 -1 2 -1 -1 12 -5 2 5 6 -4 3 1 8 -6 -2 -5 4 1 2 -25 2 -2 3
The Correct Answer and Explanation is:
To solve this, we use the formulas for the sum and product of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0:
- Sum of roots = −ba-\frac{b}{a}
- Product of roots = ca\frac{c}{a}
Let’s analyze each equation and match with the values from the answer box:
1. L: x2+x=6⇒x2+x−6=0x^2 + x = 6 \Rightarrow x^2 + x – 6 = 0
- a=1,b=1,c=−6a = 1, b = 1, c = -6
- Sum = −11=−1-\frac{1}{1} = -1, Product = −61=−6\frac{-6}{1} = -6 → Match: (-1, -6) → Letter: L
2. B: 12×2+x−6=012x^2 + x – 6 = 0
- a=12,b=1,c=−6a = 12, b = 1, c = -6
- Sum = −112-\frac{1}{12}, Product = −612=−12\frac{-6}{12} = -\frac{1}{2} → Match: (-1/12, -1/2) → Letter: B
3. G: x2−2x−2=0x^2 – 2x – 2 = 0
- a=1,b=−2,c=−2a = 1, b = -2, c = -2
- Sum = −−21=2-\frac{-2}{1} = 2, Product = −21=−2\frac{-2}{1} = -2 → Match: (2, -2) → Letter: G
4. R: 2×2+5x−25=02x^2 + 5x – 25 = 0
- a=2,b=5,c=−25a = 2, b = 5, c = -25
- Sum = −52-\frac{5}{2}, Product = −252\frac{-25}{2} → Match: (-5/2, -25/2) → Letter: R
5. A: 6×2−5x−4=06x^2 – 5x – 4 = 0
- a=6,b=−5,c=−4a = 6, b = -5, c = -4
- Sum = −−56=56-\frac{-5}{6} = \frac{5}{6}, Product = −46=−23\frac{-4}{6} = -\frac{2}{3} → Match: (5/6, -2/3) → Letter: A
6. O: 3×2−11x−4=03x^2 – 11x – 4 = 0
- a=3,b=−11,c=−4a = 3, b = -11, c = -4
- Sum = −−113=113-\frac{-11}{3} = \frac{11}{3}, Product = −43=−43\frac{-4}{3} = -\frac{4}{3} → Match: (11/3, -4/3) → Letter: O
7. E: 4x−5=−4×2⇒4×2+4x−5=04x – 5 = -4x^2 \Rightarrow 4x^2 + 4x – 5 = 0
- a=4,b=4,c=−5a = 4, b = 4, c = -5
- Sum = −44=−1-\frac{4}{4} = -1, Product = −54\frac{-5}{4} → Match: (-1, -5/4) → Letter: E
8. W: 8×2+12x+1=08x^2 + 12x + 1 = 0
- a=8,b=12,c=1a = 8, b = 12, c = 1
- Sum = −128=−32-\frac{12}{8} = -\frac{3}{2}, Product = 18\frac{1}{8} → Match: (-3/2, 1/8) → Letter: W
Final Matching:
| Sum of Roots | Product of Roots | Letter |
|---|---|---|
| 11/3 | -4/3 | O |
| 3/2 | 1/8 | W |
| -1 | -6 | L |
| 2 | -2 | G |
| -1 | -5/4 | E |
| -1/12 | -1/2 | B |
| -5/2 | -25/2 | R |
| 5/6 | -2/3 | A |
Riddle Answer:
The correct letters spell: OWL-ALGEBRA
Explanation (300 words):
This activity asks us to determine the sum and product of the roots of given quadratic equations. Using the fundamental theorem of algebra, any quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0 has roots with:
- Sum = −ba-\frac{b}{a}
- Product = ca\frac{c}{a}
We apply this to each of the 8 equations, carefully simplifying fractions. Once we calculate each pair, we match them to the corresponding values in the answer box, which gives us a letter for each pair.
For example, in the equation x2+x=6x^2 + x = 6, we rewrite it as x2+x−6=0x^2 + x – 6 = 0, so a=1,b=1,c=−6a = 1, b = 1, c = -6. This gives a sum of roots of −1-1 and product of −6-6, matching a box that reveals the letter “L”.
Repeating this process for each equation provides the full set of letters. Arranged in order of the answer box, these letters spell “OWL ALGEBRA”, which cleverly answers the riddle: “What is a smart bird’s favorite type of math?”.
The answer is: OWL-gebra!
