Given the quadratic equations:

(iii) $ 5z^2 - 39z + 28 = 0 $

(iv) $ p^2 - 15p + 56 = 0 $

Find the difference between the larger root of equation (iv) and the smaller root of equation (iii).

Question

Given the quadratic equations:

(iii) $ 5z^2 - 39z + 28 = 0 $

(iv) $ p^2 - 15p + 56 = 0 $

Find the difference between the larger root of equation (iv) and the smaller root of equation (iii).

Accepted Answer

**Solution:**

1. Consider equation (iii):
   $$5z^2 - 39z + 28 = 0$$

To find the roots, use the quadratic formula:
   $$z = \frac{39 \pm \sqrt{(-39)^2 - 4 	imes 5 	imes 28}}{2 	imes 5}$$

Calculate the discriminant:
   $$\Delta = 1521 - 560 = 961$$

$$\sqrt{961} = 31$$

So,
   $$z = \frac{39 \pm 31}{10}$$

Roots are:
   - $$z_1 = \frac{39 + 31}{10} = \frac{70}{10} = 7$$
   - $$z_2 = \frac{39 - 31}{10} = \frac{8}{10} = 0.8$$

The smaller root of equation (iii) is $$0.8$$.

2. Consider equation (iv):
   $$p^2 - 15p + 56 = 0$$

Factorizing:
   $$(p - 7)(p - 8) = 0$$

Roots are:
   - $$p = 7$$
   - $$p = 8$$

The larger root of equation (iv) is $$8$$.

3. Find the required difference:
   $$8 - 0.8 = 7.2$$

**Answer:** $7.2$

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