Given the quadratic equations derived from the variables $x$, $y$, $z$, and $p$, find the L.C.M. of their larger roots:

1. $x^2 - 5x + 6 = 0$
2. $y^2 - 9y + 20 = 0$
3. $5z^2 - 39z + 28 = 0$
4. $p^2 - 15p + 56 = 0$

What is the L.C.M. of the larger roots of $x$, $y$, $z$, and $p$?

Question

Given the quadratic equations derived from the variables $x$, $y$, $z$, and $p$, find the L.C.M. of their larger roots:

1. $x^2 - 5x + 6 = 0$
2. $y^2 - 9y + 20 = 0$
3. $5z^2 - 39z + 28 = 0$
4. $p^2 - 15p + 56 = 0$

What is the L.C.M. of the larger roots of $x$, $y$, $z$, and $p$?

Accepted Answer

**Solution:**

1. For the equation $x^2 - 5x + 6 = 0$:
   - Calculate roots using factorization:
   $$x^2 - 5x + 6 = (x - 3)(x - 2) = 0$$
   - Roots are $x = 3$ and $x = 2$.
   - Larger root is $3$.

2. For the equation $y^2 - 9y + 20 = 0$:
   - Factorize:
   $$y^2 - 9y + 20 = (y - 5)(y - 4) = 0$$
   - Roots are $y = 5$ and $y = 4$.
   - Larger root is $5$.

3. For the equation $5z^2 - 39z + 28 = 0$:
   - Use quadratic formula or factorization:
   - Factors of $5 	imes 28 = 140$ that sum to $-39$ are $-35$ and $-4$.
   - Rewrite:
   $$5z^2 - 35z - 4z + 28 = 0$$
   $$5z(z - 7) - 4(z - 7) = 0$$
   $$(5z - 4)(z - 7) = 0$$
   - Roots are $z = \frac{4}{5}$ and $z = 7$.
   - Larger root is $7$.

4. For the equation $p^2 - 15p + 56 = 0$:
   - Factorize:
   $$p^2 - 15p + 56 = (p - 7)(p - 8) = 0$$
   - Roots are $p = 7$ and $p = 8$.
   - Larger root is $8$.

5. Now, find the L.C.M. of the larger roots: $3, 5, 7,$ and $8$.

- Prime factorization:
  - $3 = 3$
  - $5 = 5$
  - $7 = 7$
  - $8 = 2^3$

- L.C.M. = $2^3 	imes 3 	imes 5 	imes 7 = 8 	imes 3 	imes 5 	imes 7 = 840$

**Answer:** $840$

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