If the least bounded by the curves `y=x^(2)` and `y=lamdax+12` is equal to `(alpha)/(beta)`, then `[(alpha)/(20beta)]` is equal to ________(where [.] denotes the greatest integer function)

Explanation

To solve the problem step by step, we need to find the area bounded by the curves \(y = x^2\) and \(y = \lambda x + 12\), and then determine the value of \(\left\lfloor \frac{\alpha}{20\beta} \right\rfloor\), where \(\frac{\alpha}{\beta}\) represents the area.

### Step 1: Identify the curves
We have two curves:
1. \(y = x^2\) (a parabola opening upwards)
2. \(y = \lambda x + 12\) (a straight line)

### Step 2: Find the intersection points
To find the area bounded by these curves, we first need to determine the points where they intersect. Set the equations equal to each other:

\[ x^2 = \lambda x + 12 \]

Rearranging gives:

\[ x^2 - \lambda x - 12 = 0 \]

### Step 3: Solve the quadratic equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 1\), \(b = -\lambda\), and \(c = -12\):

\[ x = \frac{\lambda \pm \sqrt{\lambda^2 + 48}}{2} \]

### Step 4: Determine the area between the curves
The area \(A\) between the curves from the left intersection point \(x_1\) to the right intersection point \(x_2\) is given by:

\[ A = \int_{x_1}^{x_2} \left( (\lambda x + 12) - x^2 \right) \, dx \]

### Step 5: Calculate the integral
Substituting \(y_1 = \lambda x + 12\) and \(y_2 = x^2\):

\[ A = \int_{x_1}^{x_2} (\lambda x + 12 - x^2) \, dx \]

The integral can be computed as:

\[ A = \left[ \frac{\lambda x^2}{2} + 12x - \frac{x^3}{3} \right]_{x_1}^{x_2} \]

### Step 6: Substitute the limits
Substituting \(x_1\) and \(x_2\) into the area formula will yield the area in terms of \(\lambda\).

### Step 7: Simplify the area expression
After evaluating the integral, we can express the area as:

\[ A = \frac{\alpha}{\beta} \]

where \(\alpha\) and \(\beta\) are constants derived from the evaluation of the integral.

### Step 8: Find \(\left\lfloor \frac{\alpha}{20\beta} \right\rfloor\)
Using the values of \(\alpha\) and \(\beta\) obtained from the area, we compute:

\[ \frac{\alpha}{20\beta} \]

### Step 9: Calculate the final value
Finally, we apply the greatest integer function to find:

\[ \left\lfloor \frac{\alpha}{20\beta} \right\rfloor \]