Key Concepts and Formulas

• A quadratic equation with roots $p$ and $q$ can be written as $x^2 - (p+q)x + pq = 0$.
• $(p+q)^2 = p^2 + 2pq + q^2$, which implies $p^2 + q^2 = (p+q)^2 - 2pq$.
• $(p^2 + q^2)^2 = p^4 + 2p^2q^2 + q^4$, which implies $p^4 + q^4 = (p^2 + q^2)^2 - 2(pq)^2$.

Step-by-Step Solution

Step 1: Calculate $p^2 + q^2$ in terms of $pq$

We are given that $p+q=2$. We want to express $p^2+q^2$ using $p+q$ and $pq$. Using the identity $(p+q)^2 = p^2 + 2pq + q^2$, we can rearrange to find $p^2 + q^2 = (p+q)^2 - 2pq$. Substituting $p+q=2$, we get: $p^2 + q^2 = (2)^2 - 2pq = 4 - 2pq$

Step 2: Calculate $p^4 + q^4$ in terms of $pq$

We are given that $p^4 + q^4 = 272$. We want to express this in terms of $pq$. We know that $p^4 + q^4 = (p^2 + q^2)^2 - 2(pq)^2$. Substituting $p^2 + q^2 = 4 - 2pq$ from Step 1, we get: $p^4 + q^4 = (4 - 2pq)^2 - 2(pq)^2$ Since $p^4 + q^4 = 272$, we have: $272 = (4 - 2pq)^2 - 2(pq)^2$ Expanding the square: $272 = 16 - 16pq + 4(pq)^2 - 2(pq)^2$ $272 = 16 - 16pq + 2(pq)^2$

Step 3: Solve for $pq$

Rearrange the equation to form a quadratic in $pq$: $2(pq)^2 - 16pq + 16 - 272 = 0$ $2(pq)^2 - 16pq - 256 = 0$ Divide by 2: $(pq)^2 - 8pq - 128 = 0$ Let $y = pq$. Then we have $y^2 - 8y - 128 = 0$. Factoring the quadratic, we look for two numbers that multiply to -128 and add to -8. These numbers are -16 and 8. $(y - 16)(y + 8) = 0$ So, $y = 16$ or $y = -8$. Since $p$ and $q$ are positive, $pq$ must be positive. Therefore, $pq = 16$.

Step 4: Construct the Quadratic Equation

We have $p+q = 2$ and $pq = 16$. The quadratic equation with roots $p$ and $q$ is given by $x^2 - (p+q)x + pq = 0$. Substituting the values, we have: $x^2 - (2)x + 16 = 0$ $x^2 - 2x + 16 = 0$

Common Mistakes & Tips

• Be careful with signs during algebraic manipulations. A mistake in the sign will lead to the wrong quadratic equation.
• Remember that since $p$ and $q$ are positive, their product $pq$ must also be positive. Discard any negative solutions for $pq$.
• Double-check the factoring of the quadratic equation to ensure that the roots are correct.

Summary

By using the given information $p+q=2$ and $p^4+q^4=272$, we found the value of $pq$ to be 16. Substituting these values into the general form of a quadratic equation, we obtained the equation $x^2 - 2x + 16 = 0$.

Final Answer

The final answer is \boxed{x^2 - 2x + 16 = 0}, which corresponds to option (C).