Question

solving quadratic equations using common factors quick check
which of the following quadratic equations can be solved by grouping? (1 point)
\\( x^2 - 4x - 8 = 0 \\)
\\( x^2 - 12x + 18 = 0 \\)
\\( x^2 + 8x - 22 = 0 \\)
\\( x^2 + 10x + 21 = 0 \\)

Explanation:

To determine which quadratic equation can be solved by grouping, we need to check if the equation can be factored into two binomials. For a quadratic equation \(ax^2 + bx + c = 0\) (where \(a = 1\) in all these cases), we need two numbers that multiply to \(c\) and add up to \(b\).

Step 1: Analyze \(x^2 - 4x - 8 = 0\)

We need two numbers that multiply to \(-8\) and add to \(-4\). The factors of \(-8\) are: \(\pm1, \pm2, \pm4, \pm8\). Let's check the pairs:

• \(1\) and \(-8\): \(1 + (-8) = -7\)
• \(2\) and \(-4\): \(2 + (-4) = -2\)

None of these pairs add up to \(-4\), so this equation cannot be solved by grouping.

Step 2: Analyze \(x^2 - 12x + 18 = 0\)

We need two numbers that multiply to \(18\) and add to \(-12\). The factors of \(18\) are: \(1, 2, 3, 6, 9, 18\). Since the product is positive and the sum is negative, both numbers should be negative.

• \(-1\) and \(-18\): \(-1 + (-18) = -19\)
• \(-2\) and \(-9\): \(-2 + (-9) = -11\)
• \(-3\) and \(-6\): \(-3 + (-6) = -9\)

None of these pairs add up to \(-12\), so this equation cannot be solved by grouping.

Step 3: Analyze \(x^2 + 8x - 22 = 0\)

We need two numbers that multiply to \(-22\) and add to \(8\). The factors of \(-22\) are: \(\pm1, \pm2, \pm11, \pm22\). Let's check the pairs:

• \(11\) and \(-2\): \(11 + (-2) = 9\)
• \(22\) and \(-1\): \(22 + (-1) = 21\)

None of these pairs add up to \(8\), so this equation cannot be solved by grouping.

Step 4: Analyze \(x^2 + 10x + 21 = 0\)

We need two numbers that multiply to \(21\) and add to \(10\). The factors of \(21\) are: \(1, 3, 7, 21\).

• \(3\) and \(7\): \(3 \times 7 = 21\) and \(3 + 7 = 10\)

So, we can factor the equation as \((x + 3)(x + 7) = 0\), which means it can be solved by grouping.

Answer:

\(x^2 + 10x + 21 = 0\) (the last option, i.e., the option with the equation \(x^2 + 10x + 21 = 0\))