Question
Which of the following quadratic equations can be solved by grouping
x^2+8x-22=0
x^2 -12x+ 18= 0
x^2 -4x -8 =0
x^2 +10x + 21 = 0
x^2+8x-22=0
x^2 -12x+ 18= 0
x^2 -4x -8 =0
x^2 +10x + 21 = 0
Answers
GPT-4o mini
To determine which of the given quadratic equations can be solved by grouping, we first need to identify if each equation can be factored into two binomials.
1. **\(x^2 + 8x - 22 = 0\)**:
- The factors of \(-22\) that add up to \(8\) do not exist, so this cannot be solved by grouping.
2. **\(x^2 - 12x + 18 = 0\)**:
- The factors of \(18\) that add up to \(-12\) are \(-6\) and \(-6\):
- \(x^2 - 6x - 6x + 18 = 0\), which can be grouped as:
\[
(x^2 - 6x) + (-6x + 18) = 0
\]
- This can be factored by grouping.
3. **\(x^2 - 4x - 8 = 0\)**:
- The factors of \(-8\) that add up to \(-4\) do not exist, so this cannot be solved by grouping.
4. **\(x^2 + 10x + 21 = 0\)**:
- The factors of \(21\) that add up to \(10\) are \(3\) and \(7\):
- \(x^2 + 3x + 7x + 21 = 0\), which can be grouped as:
\[
(x^2 + 3x) + (7x + 21) = 0
\]
- This can be factored by grouping.
In conclusion, the quadratic equations that can be solved by grouping are:
- \(x^2 - 12x + 18 = 0\)
- \(x^2 + 10x + 21 = 0\)
1. **\(x^2 + 8x - 22 = 0\)**:
- The factors of \(-22\) that add up to \(8\) do not exist, so this cannot be solved by grouping.
2. **\(x^2 - 12x + 18 = 0\)**:
- The factors of \(18\) that add up to \(-12\) are \(-6\) and \(-6\):
- \(x^2 - 6x - 6x + 18 = 0\), which can be grouped as:
\[
(x^2 - 6x) + (-6x + 18) = 0
\]
- This can be factored by grouping.
3. **\(x^2 - 4x - 8 = 0\)**:
- The factors of \(-8\) that add up to \(-4\) do not exist, so this cannot be solved by grouping.
4. **\(x^2 + 10x + 21 = 0\)**:
- The factors of \(21\) that add up to \(10\) are \(3\) and \(7\):
- \(x^2 + 3x + 7x + 21 = 0\), which can be grouped as:
\[
(x^2 + 3x) + (7x + 21) = 0
\]
- This can be factored by grouping.
In conclusion, the quadratic equations that can be solved by grouping are:
- \(x^2 - 12x + 18 = 0\)
- \(x^2 + 10x + 21 = 0\)