\(\mathrm{x(x + 1) = 3}\)What values of x satisfy the equation above?

1. SIMPLIFY the equation by expanding

• Given: \(\mathrm{x(x + 1) = 3}\)
• Expand the left side using distributive property: \(\mathrm{x^2 + x = 3}\)

2. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{x^2 + x - 3 = 0}\)
• This is now in the form \(\mathrm{ax^2 + bx + c = 0}\) where \(\mathrm{a = 1, b = 1, c = -3}\)

3. INFER the solution method

• Check if this factors easily: we need two numbers that multiply to -3 and add to 1
• Since no simple factors work, use the quadratic formula

4. SIMPLIFY using the quadratic formula

• Formula: \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\)
• Substitute values: \(\mathrm{x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-3)}}{2(1)}}\)
• Calculate discriminant: \(\mathrm{x = \frac{-1 \pm \sqrt{1 + 12}}{2}}\)
• Final form: \(\mathrm{x = \frac{-1 \pm \sqrt{13}}{2}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when rearranging to standard form

Students often write \(\mathrm{x^2 - x - 3 = 0}\) instead of \(\mathrm{x^2 + x - 3 = 0}\), mixing up the sign of the x term during rearrangement. When they apply the quadratic formula to this incorrect equation, they get \(\mathrm{x = \frac{1 \pm \sqrt{13}}{2}}\).

This leads them to select Choice D (\(\mathrm{\frac{1+\sqrt{13}}{2}}\) and \(\mathrm{\frac{1-\sqrt{13}}{2}}\))

Second Most Common Error:

Poor INFER reasoning: Attempting to force factoring instead of using quadratic formula

Students may waste time trying to factor \(\mathrm{x^2 + x - 3 = 0}\) or incorrectly assume it factors, leading to wrong integer solutions. When factoring fails, they may guess or attempt incorrect factorizations.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires careful algebraic manipulation and recognition of when the quadratic formula is necessary. The key challenge is maintaining accuracy through multiple algebraic steps while making the strategic decision to use the quadratic formula rather than attempting to factor.