Solve the Quadratic Equation for X: ax + a² = (a + x)² + 3

Look at the equation below and express X in terms of a, given that a<3.

$ax+a^2=(a+x)^2+3$

To find $x$ in terms of $a$ from the equation $ax + a^2 = (a + x)^2 + 3$, follow these steps:

• Step 1: Expand the right-hand side:
  $(a + x)^2 = a^2 + 2ax + x^2$
• Step 2: Substitute and simplify the equation:
  $ax + a^2 = a^2 + 2ax + x^2 + 3$
• Step 3: Rearrange the terms:
  $ax + a^2 - a^2 = 2ax + x^2 + 3$
  $ax = 2ax + x^2 + 3$
• Step 4: Move all terms to one side of the equation to form a quadratic:
  $0 = x^2 + ax + 3$
• Step 5: Analyze the quadratic equation $x^2 + ax + 3 = 0$:
  Calculate the discriminant: $\Delta = a^2 - 4 \cdot 1 \cdot 3 = a^2 - 12$.
• Step 6: Consider the condition for real solutions ($\Delta \geq 0$):
  Since $a^2 - 12 < 0$ for $a < 3$, the discriminant is negative for values of $a < 3$. This means the quadratic equation has no real solutions.

Therefore, for the given condition $a < 3$, there is no solution for $x$.