### Part (ह): Solve the Equation [tex]\(9^x - 4 \cdot 3^x + 3 = 0\)[/tex]
We start with the given equation:
[tex]\[ 9^x - 4 \cdot 3^x + 3 = 0 \][/tex]
First, notice that [tex]\(9^x\)[/tex] can be written as [tex]\((3^2)^x = (3^x)^2\)[/tex]. Let [tex]\(y = 3^x\)[/tex]. Substituting [tex]\(y\)[/tex] into the equation, we get:
[tex]\[ (3^x)^2 - 4 \cdot 3^x + 3 = 0 \][/tex]
[tex]\[ y^2 - 4y + 3 = 0 \][/tex]
Now, we solve this quadratic equation in [tex]\(y\)[/tex]. The standard form of a quadratic equation is [tex]\(ay^2 + by + c = 0\)[/tex], where in our case [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = 3\)[/tex]. We solve using the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} \][/tex]
[tex]\[ y = \frac{4 \pm \sqrt{16 - 12}}{2} \][/tex]
[tex]\[ y = \frac{4 \pm \sqrt{4}}{2} \][/tex]
[tex]\[ y = \frac{4 \pm 2}{2} \][/tex]
This gives us two solutions for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{4 + 2}{2} = 3 \][/tex]
[tex]\[ y = \frac{4 - 2}{2} = 1 \][/tex]
Recall that [tex]\(y = 3^x\)[/tex]. Thus:
[tex]\[ 3^x = 3 \implies x = 1 \][/tex]
[tex]\[ 3^x = 1 \implies x = 0 \][/tex]
Therefore, the solutions to the equation [tex]\(9^x - 4 \cdot 3^x + 3 = 0\)[/tex] are:
[tex]\[ x = 0 \quad \text{and} \quad x = 1 \][/tex]
### Part (ख): Simplify the Expression [tex]\(\frac{a^2 + b}{a^2 - b} + \frac{a^2 - b}{a^2 + b} - \frac{a^4 + b^2}{a^4 - b^2}\)[/tex]
We simplify the given expression step-by-step:
[tex]\[ \frac{a^2 + b}{a^2 - b} + \frac{a^2 - b}{a^2 + b} - \frac{a^4 + b^2}{a^4 - b^2} \][/tex]
First, recognize that the middle term [tex]\(\frac{a^2 - b}{a^2 + b}\)[/tex] is the reciprocal of the first term [tex]\(\frac{a^2 + b}{a^2 - b}\)[/tex]. Let's combine the first two terms:
[tex]\[ \frac{a^2 + b}{a^2 - b} + \frac{a^2 - b}{a^2 + b} \][/tex]
To add these fractions, we need a common denominator, which is [tex]\((a^2 - b)(a^2 + b)\)[/tex]:
[tex]\[ \frac{(a^2 + b)^2 + (a^2 - b)^2}{(a^2 - b)(a^2 + b)} \][/tex]
Simplify the numerator:
[tex]\[ (a^2 + b)^2 = a^4 + 2a^2b + b^2 \][/tex]
[tex]\[ (a^2 - b)^2 = a^4 - 2a^2b + b^2 \][/tex]
Therefore, we have:
[tex]\[ \frac{a^4 + 2a^2b + b^2 + a^4 - 2a^2b + b^2}{(a^2 - b)(a^2 + b)} = \frac{2a^4 + 2b^2}{a^4 - b^2} \][/tex]
[tex]\[ = \frac{2(a^4 + b^2)}{a^4 - b^2} \][/tex]
Now, we subtract the third term:
[tex]\[ \frac{2(a^4 + b^2)}{a^4 - b^2} - \frac{a^4 + b^2}{a^4 - b^2} \][/tex]
Since they have the same denominator, we combine the numerators:
[tex]\[ \frac{2(a^4 + b^2) - (a^4 + b^2)}{a^4 - b^2} \][/tex]
[tex]\[ = \frac{2a^4 + 2b^2 - a^4 - b^2}{a^4 - b^2} \][/tex]
[tex]\[ = \frac{a^4 + b^2}{a^4 - b^2} \][/tex]
Thus, the simplified expression is:
[tex]\[ \frac{a^4 + b^2}{a^4 - b^2} \][/tex]
