Question:

Step 1: Identify the equation satisfied by $a$ and $b$ Given $a^2 = 5a - 3$ and $b^2 = 5b - 3$. Rearranging these equations, we obtain $a^2 - 5a + 3 = 0$ and $b^2 - 5b + 3 = 0$. Since $a \neq b$, $a$ and $b$ are the distinct roots of the quadratic equation: $$ x^2 - 5x + 3 = 0 $$ Step 2: Apply Vieta's formulas For the quadratic equation $x^2 - 5x + 3 = 0$, Vieta's formulas state: The sum of the roots is $a + b = -\frac{(-5)}{1} = 5$. The product of the roots is $ab = \frac{3}{1} = 3$. Step 3: Find the sum of the new roots Let the new roots be $\alpha = \frac{a}{b}$ and $\beta = \frac{b}{a}$. The sum of the new roots is: $$ \alpha + \beta = \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} $$ To find $a^2 + b^2$, we use the identity $a^2 + b^2 = (a+b)^2 - 2ab$. Substituting the values from Step 2: $$ a^2 + b^2 = (5)^2 - 2(3) = 25 - 6 = 19 $$ Therefore, the sum of the new roots is: $$ \alpha + \beta = \frac{19}{3} $$ Step 4: Find the product of the new roots The product of the new roots is: $$ \alpha \cdot \beta = \left(\frac{a}{b}\right) \cdot \left(\frac{b}{a}\right) = 1 $$ Step 5: Form the quadratic equation A quadratic equation with roots $\alpha$ and $\beta$ is given by $x^2 - (\alpha + \beta)x + \alpha\beta = 0$. Substituting the values from Step 3 and Step 4: $$ x^2 - \left(\frac{19}{3}\right)x + 1 = 0 $$ To eliminate the fraction, multiply the entire equation by 3: $$ 3x^2 - 19x + 3 = 0 $$