1. Find possible values of the constant p if the expression Px^2 - 12x + 4p is a perfect square.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the questions:
1. Find possible values of the constant $p$ if the expression $Px^2 - 12x + 4p$ is a perfect square.
For a quadratic expression $Ax^2 + Bx + C$ to be a perfect square, its discriminant must be zero, i.e., $B^2 - 4AC = 0$.
In this expression, $A = P$, $B = -12$, and $C = 4p$.
Step 1: Set the discriminant to zero.
$$(-12)^2 - 4(P)(4p) = 0$$
Step 2: Simplify the equation.
$$144 - 16P^2
