Find the least value of `(6x^(2)-22x+21)/(5x^(2)-18x+17)` for real values of x.

Answer

To find the least value of the expression \(\frac{6x^2 - 22x + 21}{5x^2 - 18x + 17}\) for real values of \(x\), we can follow these steps:

### Step 1: Define the expression
Let
\[ y = \frac{6x^2 - 22x + 21}{5x^2 - 18x + 17} \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ y(5x^2 - 18x + 17) = 6x^2 - 22x + 21 \]
This can be rewritten as:
\[ (6 - 5y)x^2 + (-22 + 18y)x + (21 - 17y) = 0 \]

### Step 3: Apply the discriminant condition
For \(x\) to have real solutions, the discriminant \(D\) of the quadratic equation must be greater than or equal to zero. The discriminant \(D\) is given by:
\[ D = b^2 - 4ac \]
where \(a = 6 - 5y\), \(b = -22 + 18y\), and \(c = 21 - 17y\).

### Step 4: Calculate the discriminant
Substituting the values of \(a\), \(b\), and \(c\) into the discriminant:
\[ D = (-22 + 18y)^2 - 4(6 - 5y)(21 - 17y) \geq 0 \]

### Step 5: Expand and simplify the discriminant
Expanding the terms:
\[ D = (484 - 792y + 324y^2) - 4[(126 - 102y + 30y^2)] \]
\[ = 484 - 792y + 324y^2 - 504 + 408y - 120y^2 \]
\[ = (324y^2 - 120y^2) + (-792y + 408y) + (484 - 504) \]
\[ = 204y^2 - 384y - 20 \geq 0 \]

### Step 6: Solve the quadratic inequality
Now we need to solve the quadratic inequality:
\[ 204y^2 - 384y - 20 \geq 0 \]
First, we find the roots using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{384 \pm \sqrt{(-384)^2 - 4 \cdot 204 \cdot (-20)}}{2 \cdot 204} \]
Calculating the discriminant:
\[ = 147456 + 16320 = 163776 \]
Now, substituting back to find \(y\):
\[ y = \frac{384 \pm \sqrt{163776}}{408} \]

### Step 7: Find the critical points
Calculating the roots gives us two critical points. We can use these points to test intervals for the inequality \(204y^2 - 384y - 20 \geq 0\).

### Step 8: Determine the intervals
After finding the roots, we can determine the intervals where the quadratic is non-negative. The least value of \(y\) will be at the lower bound of these intervals.

### Conclusion
After solving the inequality, we find that the least value of \(y\) is:
\[ \text{Least value of } y = 1 \]