In which of the following equations is the difference between the larger and smaller root exactly 1?

![Screenshot 2025-10-15 184855.png](https://res.cloudinary.com/aptidude/image/upload/v1760534485/aptidude_questions/e9enilcsevdxy6n3pblg.png)

Question

In which of the following equations is the difference between the larger and smaller root exactly 1?

![Screenshot 2025-10-15 184855.png](https://res.cloudinary.com/aptidude/image/upload/v1760534485/aptidude_questions/e9enilcsevdxy6n3pblg.png)

Accepted Answer

**Solution:**

1. For equation (i):
   $$ x^2 - 3x - \sqrt{4x^2} = -6 $$
   Since $\sqrt{4x^2} = 2x$, rewrite as:
   $$ x^2 - 3x - 2x = -6 \implies x^2 - 5x + 6 = 0 $$
   Factorizing or using quadratic formula:
   $$ x = 3, 2 $$
   Difference between roots = $3 - 2 = 1$

2. For equation (ii):
   $$ y^2 - \sqrt{81y^2} = -20 $$
   Since $\sqrt{81y^2} = 9y$, rewrite as:
   $$ y^2 - 9y = -20 \implies y^2 - 9y + 20 = 0 $$
   Roots are:
   $$ y = 5, 4 $$
   Difference = $5 - 4 = 1$

3. For equation (iii):
   $$ \frac{z^2 \sqrt{625z^6}}{5z^3} + 28 = 39z $$
   Note that $\sqrt{625z^6} = 25z^3$, so:
   $$ \frac{z^2 \times 25z^3}{5z^3} + 28 = 39z \implies 5z^2 + 28 = 39z $$
   Rearranged:
   $$ 5z^2 - 39z + 28 = 0 $$
   Roots:
   $$ z = 7, \frac{4}{5} = 0.8 $$
   Difference = $7 - 0.8 = 6.2 \neq 1$

4. For equation (iv):
   $$ p^2 - 15p = -56 \implies p^2 - 15p + 56 = 0 $$
   Roots:
   $$ p = 7, 8 $$
   Difference = $8 - 7 = 1$

**Answer:** Only equations (i), (ii), and (iv) have roots whose difference is exactly 1.

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