2x^2 + 12x - 35 = 0What is the product of the solutions to the given equation?

1. TRANSLATE the problem information

• Given equation: \(2\mathrm{x}^2 + 12\mathrm{x} - 35 = 0\)
• Need to find: Product of the solutions
• What this tells us: We have a quadratic equation in standard form

2. INFER the most efficient approach

• Instead of solving for both roots individually, we can use Vieta's formulas
• For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), the product of solutions equals \(\frac{\mathrm{c}}{\mathrm{a}}\)
• This saves time and reduces calculation errors

3. TRANSLATE to identify coefficients

• From \(2\mathrm{x}^2 + 12\mathrm{x} - 35 = 0\):
  • \(\mathrm{a} = 2\) (coefficient of x²)
  • \(\mathrm{b} = 12\) (coefficient of x)
  • \(\mathrm{c} = -35\) (constant term)

4. SIMPLIFY using Vieta's formula

• Product of solutions = \(\frac{\mathrm{c}}{\mathrm{a}}\)
• Product = \(\frac{-35}{2}\)

Answer: (A) \(\frac{-35}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize they can use Vieta's formulas and instead try to solve the quadratic completely by factoring or using the quadratic formula.

They might attempt to factor \(2\mathrm{x}^2 + 12\mathrm{x} - 35 = 0\), struggle with the arithmetic, make errors in the process, and either get frustrated or compute incorrect roots like \(\mathrm{x} = \frac{5}{2}\) and \(\mathrm{x} = -7\). Then they multiply these wrong values to get \(\frac{-35}{4}\), leading them to select an incorrect answer choice or become confused.

Second Most Common Error:

Poor TRANSLATE execution: Students misidentify the coefficients, particularly confusing the signs or forgetting that \(\mathrm{c} = -35\) (not +35).

If they think \(\mathrm{c} = 35\), they calculate the product as \(\frac{35}{2}\) and select Choice (D) (\(\frac{35}{2}\)), missing the crucial negative sign in the original equation.

The Bottom Line:

This problem tests whether students know the elegant shortcut of Vieta's formulas versus getting bogged down in unnecessary computation. The key insight is recognizing that quadratic relationships give us direct access to root properties without finding the actual roots.