Question

Which of the following quadratic equations is *not* solvable by grouping?

x^2 - 2x + 1 = 0
x^2 - 12x + 35 = 0
2x^2 - 2x - 10 = 0
2x^2 + 14x + 12 =0

Answer 1

To determine which quadratic equation is *not* solvable by grouping, we should look at the possibility of expressing each quadratic in a form that can be factored into two binomials. The key characteristic of equations solvable by grouping is that we can rewrite them in such a way that we can factor out common terms.

1. **Equation:** \( x^2 - 2x + 1 = 0 \)

This factors as \( (x - 1)^2 = 0 \). It can be solved by grouping, as there are common factors.

2. **Equation:** \( x^2 - 12x + 35 = 0 \)

This can be factored as \( (x - 5)(x - 7) = 0 \). It can also be solved by grouping.

3. **Equation:** \( 2x^2 - 2x - 10 = 0 \)

We can factor out a common factor of 2:
\( 2(x^2 - x - 5) = 0 \).
Then \( x^2 - x - 5 = 0 \) does not factor nicely, nor does it yield simple factors. While this can still be solved using the quadratic formula, it’s not easily factored by grouping.

4. **Equation:** \( 2x^2 + 14x + 12 = 0 \)

This can be factored by taking out a common factor of 2 first:
\( 2(x^2 + 7x + 6) = 0 \).
Then \( x^2 + 7x + 6 \) factors as \( (x + 6)(x + 1) = 0 \).

Thus, we see that:

- The first equation is solvable by grouping.
- The second equation is solvable by grouping.
- The fourth equation is solvable by grouping.
- The third equation, \( 2x^2 - 2x - 10 = 0 \), does not factor easily and can't be solved by grouping.

Therefore, the quadratic equation that is *not* solvable by grouping is:

**2x^2 - 2x - 10 = 0**.