Which of the following quadratic equations can be solved by grouping?

Question

$x^{2}+10x+21=0$
$x^{2}-12x+18=0$
$x^{2}+8x-22=0$
$x^{2}-4x-8=0$

Answer 1

To determine which of the quadratic equations can be solved by grouping, we can look for equations that can be factored into two binomials, where the middle term can be split into two terms that can be grouped.

1. Equation: \(x^{2}+10x+21=0\)

   To factor:

   • We need two numbers that multiply to \(21\) and add to \(10\).
   • The numbers \(3\) and \(7\) work since \(3 + 7 = 10\) and \(3 \times 7 = 21\).
   • Factoring: \((x + 3)(x + 7) = 0\)

2. Equation: \(x^{2}-12x+18=0\)

   To factor:

   • We need two numbers that multiply to \(18\) and add to \(-12\).
   • The only integer pairs \((1, 18)\), \((2, 9)\), and \((3, 6)\) do not sum to \(-12\). Hence, it cannot be factored simply.

3. Equation: \(x^{2}+8x-22=0\)

   To factor:

   • We need two numbers that multiply to \(-22\) and add to \(8\).
   • The pairs are \((-2, 11)\) or \((2, -11)\). Neither pair sums to \(8\), so it cannot be factored simply.

4. Equation: \(x^{2}-4x-8=0\)

   To factor:

   • We need two numbers that multiply to \(-8\) and add to \(-4\).
   • The pairs are \((2, -4)\) or \((-2, 4)\) does not work. Hence, it cannot be factored simply.

From the analysis, the equation that can be solved by grouping is:

\[ \boxed{x^{2}+10x+21=0} \]