2x^2+22x+60 fill in the blanks below

Question

Part A. The GCF is = ____
The quadratic with the factored out the GCF is ____
a = ___. b = _____ c = ___-
Part B (use the quadratic formula
x = ____________
Part C solve for x
x = ___ lesser number
x = _____ greater number

Answer 1

Let’s solve the problem step by step.

Part A

1. Finding the GCF (Greatest Common Factor): The expression is \( 2x^2 + 22x + 60 \). We can find the GCF of the coefficients (2, 22, and 60). The GCF is \( 2 \).

2. Factoring out the GCF: Factoring out \( 2 \): \[ 2(x^2 + 11x + 30) \]

   So, the quadratic with the GCF factored out is \( 2(x^2 + 11x + 30) \).

3. Identifying a, b, and c: In the expression \( x^2 + 11x + 30 \):

   • \( a = 1 \)
   • \( b = 11 \)
   • \( c = 30 \)

Filling in Part A:

• The GCF is = 2
• The quadratic with the factored out the GCF is x² + 11x + 30
• \( a = 1 \)
• \( b = 11 \)
• \( c = 30 \)

Part B

Using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \):

1. Calculate the discriminant: \[ b^2 - 4ac = 11^2 - 4(1)(30) = 121 - 120 = 1 \]
2. Now apply the quadratic formula: \[ x = \frac{{-11 \pm \sqrt{1}}}{2(1)} = \frac{{-11 \pm 1}}{2} \]

Filling in Part B:

• \( x = \frac{{-11 \pm 1}}{2} \)

Part C

Now we need to solve for x using the results from Part B.

1. Calculating the two potential values for \( x \):
   • For the positive case: \[ x = \frac{{-11 + 1}}{2} = \frac{{-10}}{2} = -5 \]
   • For the negative case: \[ x = \frac{{-11 - 1}}{2} = \frac{{-12}}{2} = -6 \]

Filling in Part C:

• \( x = -6 \) lesser number
• \( x = -5 \) greater number

Final Summary:

• Part A:

  • The GCF is = 2
  • The quadratic with the factored out the GCF is x² + 11x + 30
  • \( a = 1 \)
  • \( b = 11 \)
  • \( c = 30 \)

• Part B:

  • \( x = \frac{{-11 \pm 1}}{2} \)

• Part C:

  • \( x = -6 \) lesser number
  • \( x = -5 \) greater number