In this problem, the goal is to match each quadratic equation with its corresponding letter and the set of zeros, then use the indicated letter to form a hidden message.

Step 1: Identify the zeros of each quadratic function

We need to factor each quadratic function (or use the quadratic formula when necessary) to find its zeros.

1. Y: $y = 4x^2 - 25$
   This is a difference of squares:
   $4x^2 - 25 = (2x - 5)(2x + 5)$
   Zeros: $x = \pm \frac{5}{2}$

2. R: $y = x^2 - 9$
   Another difference of squares:
   $x^2 - 9 = (x - 3)(x + 3)$
   Zeros: $x = \pm 3$

3. V: $y = 9x^2 - 16$
   Also a difference of squares:
   $9x^2 - 16 = (3x - 4)(3x + 4)$
   Zeros: $x = \pm \frac{4}{3}$

4. G: $y = x^2 + 6x + 9$
   This is a perfect square trinomial:
   $x^2 + 6x + 9 = (x + 3)^2$
   Zeros: $x = -3$ (double root)

5. U: $y = x^2 - 4x - 21$
   Factorization:
   $x^2 - 4x - 21 = (x - 7)(x + 3)$
   Zeros: $x = 7, -3$

6. S: $y = 6x^2 + 5x - 4$
   Use the quadratic formula:
   $x = \frac{-5 \pm \sqrt{5^2 - 4(6)(-4)}}{2(6)} = \frac{-5 \pm \sqrt{25 + 96}}{12} = \frac{-5 \pm \sqrt{121}}{12} = \frac{-5 \pm 11}{12}$
   Zeros: $x = \frac{1}{2}, -\frac{2}{3}$

7. E: $y = x^2 - 5x - 36$
   Factorization:
   $x^2 - 5x - 36 = (x - 9)(x + 4)$
   Zeros: $x = 9, -4$

8. L: $y = x^2 - x - 20$
   Factorization:
   $x^2 - x - 20 = (x - 5)(x + 4)$
   Zeros: $x = 5, -4$

9. D: $y = 2x^2 + x - 3$
   Use the quadratic formula:
   $x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-3)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 24}}{4} = \frac{-1 \pm \sqrt{25}}{4} = \frac{-1 \pm 5}{4}$
   Zeros: $x = \frac{1}{2}, -\frac{3}{2}$

10. O: $y = 6x^2 - 7x + 2$
    Use the quadratic formula:
    $x = \frac{7 \pm \sqrt{(-7)^2 - 4(6)(2)}}{2(6)} = \frac{7 \pm \sqrt{49 - 48}}{12} = \frac{7 \pm 1}{12}$
    Zeros: $x = \frac{2}{3}, \frac{1}{2}$

Step 2: Match the zeros to the corresponding functions

Now that we have the zeros, we can match each set of zeros to the correct function and fill in the letters.

• { -3, -3 } → G (since $G: x = -3$ (double root))
• { 2/3, 1/2 } → O (since $O: x = \frac{2}{3}, \frac{1}{2}$)
• { -2/3, 1 } → S (since $S: x = -\frac{2}{3}, \frac{1}{2}$)

Next line:

• { 5, -4 } → L (since $L: x = 5, -4$)
• { 2/3, 1/2 } → O (same as above)
• { 4/3, -4/3 } → V (since $V: x = \frac{4}{3}, -\frac{4}{3}$)
• { 9, -4 } → E (since $E: x = 9, -4$)
• { 4/3, 1 } → S (since $S: x = -\frac{2}{3}, \frac{1}{2}$)

Final line:

• { 5/2, -5/2 } → Y (since $Y: x = \pm \frac{5}{2}$)
• { 2/3, 1/2 } → O (same as above)
• { 7, -3 } → U (since $U: x = 7, -3$)

Step 3: Form the hidden message

The message can be read by the letters corresponding to each set of zeros. The letters are:

• G O S
• L O V E S
• Y O U

The hidden message is: "GOS LOVES YOU".

Let me know if you need more details or explanations! Here are 5 related questions:

1. How do you factor quadratic equations using the difference of squares method?
2. What are the steps to apply the quadratic formula?
3. What are double roots, and how do they appear in quadratic equations?
4. How can you determine the nature of the roots (real, complex) based on the discriminant?
5. How do you factor quadratic trinomials?

Tip: Always check if a quadratic can be factored easily before applying the quadratic formula—it can save time!