x^2 + 5x - 3 = x^2 + 8x - 15 How many solutions does the given equation have?...

1. INFER the key insight

• Given equation: \(\mathrm{x^2 + 5x - 3 = x^2 + 8x - 15}\)
• Key insight: Both sides have \(\mathrm{x^2}\) terms that we can eliminate
• Strategy: Subtract \(\mathrm{x^2}\) from both sides to reveal the true nature of this equation

2. SIMPLIFY by eliminating the quadratic terms

• Subtract \(\mathrm{x^2}\) from both sides:

\(\mathrm{x^2 + 5x - 3 = x^2 + 8x - 15}\)

\(\mathrm{5x - 3 = 8x - 15}\)

• Now we have a linear equation instead of quadratic!

3. SIMPLIFY to isolate the variable

• Subtract 5x from both sides:

\(\mathrm{5x - 3 = 8x - 15}\)

\(\mathrm{-3 = 3x - 15}\)

• Add 15 to both sides:

\(\mathrm{-3 = 3x - 15}\)

\(\mathrm{12 = 3x}\)

• Divide both sides by 3:

\(\mathrm{x = 4}\)

4. INFER the final answer

• We found exactly one value: \(\mathrm{x = 4}\)
• Linear equations typically have exactly one solution (unless they're inconsistent or identities)

Answer: B (Exactly one)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see \(\mathrm{x^2}\) terms and immediately think "quadratic equation with two solutions" without recognizing they can eliminate the quadratic terms.

They may try to rearrange into standard form (\(\mathrm{x^2 + 5x - 3 = x^2 + 8x - 15}\) → \(\mathrm{0 = 3x - 12}\)) but then get confused about why there's no \(\mathrm{x^2}\) term left, leading to guessing or incorrectly assuming there should be two solutions.

This may lead them to select Choice C (Exactly two).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the algebraic steps, such as:

• Sign errors when subtracting terms
• Incorrect addition/subtraction of constants
• Division mistakes

For example, getting \(\mathrm{-3 = 3x - 15}\) but then adding 15 incorrectly to get \(\mathrm{18 = 3x}\), leading to \(\mathrm{x = 6}\) instead of \(\mathrm{x = 4}\). When they can't verify this works in the original equation, they may assume there are no solutions.

This may lead them to select Choice A (Zero).

The Bottom Line:

This problem tests whether students can recognize that identical terms on both sides of an equation can be eliminated, transforming what appears to be a quadratic into a simple linear equation. The key insight is that not every equation with \(\mathrm{x^2}\) terms is truly quadratic.