\(\mathrm{x(x - 3) + 5 = x(4 - 2x) + 3}\) What is the product of the solutions to the...

1. SIMPLIFY the equation by expanding both sides

• Given: \(\mathrm{x(x - 3) + 5 = x(4 - 2x) + 3}\)
• Expand left side: \(\mathrm{x(x - 3) + 5 = x^2 - 3x + 5}\)
• Expand right side: \(\mathrm{x(4 - 2x) + 3 = 4x - 2x^2 + 3}\)
• Equation becomes: \(\mathrm{x^2 - 3x + 5 = 4x - 2x^2 + 3}\)

2. SIMPLIFY by rearranging to standard quadratic form

• Move all terms to left side: \(\mathrm{x^2 - 3x + 5 - 4x + 2x^2 - 3 = 0}\)
• Combine like terms carefully:
  • x² terms: \(\mathrm{x^2 + 2x^2 = 3x^2}\)
  • x terms: \(\mathrm{-3x - 4x = -7x}\)
  • Constants: \(\mathrm{5 - 3 = 2}\)
• Standard form: \(\mathrm{3x^2 - 7x + 2 = 0}\)

3. INFER the most efficient approach

• The problem asks for the product of solutions, not the individual solutions
• For any quadratic \(\mathrm{ax^2 + bx + c = 0}\), the product of roots = \(\mathrm{\frac{c}{a}}\)
• This saves time compared to solving completely

4. SIMPLIFY using the product formula

• From \(\mathrm{3x^2 - 7x + 2 = 0}\): \(\mathrm{a = 3, b = -7, c = 2}\)
• Product of solutions = \(\mathrm{\frac{c}{a} = \frac{2}{3}}\)

Answer: \(\mathrm{\frac{2}{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when expanding expressions or combining like terms.

Common mistakes include getting the wrong sign when expanding \(\mathrm{x(4 - 2x)}\) or incorrectly combining the x terms when rearranging. For example, they might get \(\mathrm{3x^2 - 5x + 2 = 0}\) instead of \(\mathrm{3x^2 - 7x + 2 = 0}\), leading to a product of \(\mathrm{\frac{2}{3}}\) calculated from wrong coefficients, or they abandon the systematic approach and guess.

Second Most Common Error:

Missing conceptual knowledge: Students don't know Vieta's formulas for the product of roots.

Without knowing that product = \(\mathrm{\frac{c}{a}}\), they attempt to solve the entire quadratic equation using the quadratic formula or factoring. While this approach works, it's more time-consuming and creates more opportunities for calculation errors. They may get correct individual solutions but make arithmetic mistakes when multiplying them together.

The Bottom Line:

This problem rewards students who recognize the efficient path (product formula) and execute algebraic manipulation accurately. The combination of multi-step simplification with strategic thinking makes it challenging for students weak in either area.