Solving Quadratic Equations Using Common Factors Quick Check 5 of 55 of 5 Items Question Which of the following quadratic equations can be solved by grouping?(1 point) Responses x^2+10x+21=0, x squared plus 10 x plus 21 equals 0, x^2+8x−22=0, x squared plus 8 x minus 22 equals 0 x2−4x−8=0, x squared minus 4 x minus 8 equals 0 x2−12x+18=0 x squared minus 12 x plus 18 equals 0

Answer 1

To determine which of the given quadratic equations can be solved by grouping, we need to look for a way to factor each equation into pairs that can be grouped together.

Let's analyze each equation:

1. \(x^2 + 10x + 21 = 0\)

   • Factors as \((x + 3)(x + 7) = 0\).
   • This equation cannot be solved by grouping as it does not require grouping.

2. \(x^2 + 8x - 22 = 0\)

   • Look for two numbers that multiply to \(-22\) and add to \(8\). The numbers \(11\) and \(-2\) work. So it can be factored to \((x + 11)(x - 2) = 0\).
   • However, it's again not a case for grouping.

3. \(x^2 - 4x - 8 = 0\)

   • Look for two numbers that multiply to \(-8\) and add to \(-4\). The numbers do not easily factor, suggesting it is not a grouping case.

4. \(x^2 - 12x + 18 = 0\)

   • Look for two numbers that multiply to \(18\) and add to \(-12\). The numbers would be difficult to find; hence it does not suit grouping well.

To perform grouping effectively, we typically look for a polynomial that can be rearranged into two parts that share common factors. None of these quadratic equations seem to fit easily into a grouping method directly, as they either do not factor nicely or do so in a way that does not involve grouping.

Nevertheless, if we had to choose the one most likely aligning with the grouping method (with a small potential of manipulating into two grouped factors), it would probably be this one:

\(x^2 - 4x - 8 = 0\)

It is the easiest case where the expression could lead to further calculations to get simpler factors when possible, such as completing the square or similar method, should be considered for that analysis.

So, the correct answer using guidance on common factors while assessing for potential grouping would lead me to tentatively give:

\(x^2 - 4x - 8 = 0\) as closest to suit consideration for solving through grouping.

However, note that truly effective grouping methods would be more apparent in higher degree polynomials.

Answer 2

x squared +10+21=0 is the correct answer

Answer 3

It seems there was some confusion in the analysis of the provided equation \(x^2 + 10x + 21 = 0\). Let's revisit it:

The equation \(x^2 + 10x + 21 = 0\) can be factored as follows:

1. Look for two numbers that multiply to \(21\) (the constant term) and add to \(10\) (the coefficient of the linear term).
2. The numbers \(3\) and \(7\) fit here since \(3 \times 7 = 21\) and \(3 + 7 = 10\).

Thus, we can factor the equation:

\[ (x + 3)(x + 7) = 0 \]

This shows that it indeed can be solved by factoring, but it does not specifically involve grouping, which is typically used when you have four terms and can rearrange or combine pairs.

Summary:

• The quadratic equation \(x^2 + 10x + 21 = 0\) can be solved via factoring.
• Grouping is usually applied to polynomials with four terms or for certain cases where one can rearrange terms strategically.

If you’re specifically looking for quadratic equations that can be approached using grouping (like when you have something like \(ax^2 + bx + c\) rearranged or configured ideally), then yes, the \(x^2 + 10x + 21 = 0\) can yield a solution through standard factoring methods.

Apologies for any confusion in the prior response! Thank you for your patience.